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Contents
Tight relative t-designs on two shells in hypercubes, and Hahn and Hermite
polynomials
Eiichi Bannai, Etsuko Bannai, Hajime Tanaka, Yan Zhu . . . . . . . . . . . 163
Two-distance transitive normal Cayley graphs
Jun-Jie Huang, Yan-Quan Feng, Jin-Xin Zhou . . . . . . . . . . . . . . . . 207
LDPC codes from cubic semisymmetric graphs
Dean Crnković, Sanja Rukavina, Marina Šimac . . . . . . . . . . . . . . . 217
Paired domination stability in graphs
Aleksandra Gorzkowska, Michael A. Henning, Monika Pilśniak,
Elżbieta Tumidajewicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Notes on weak-odd edge colorings of digraphs
César Hernández-Cruz, Mirko Petruševski, Riste Škrekovski . . . . . . . . 249
A-trails of embedded graphs and twisted duals
Qi Yan, Xian’an Jin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Generalised dihedral CI-groups
Ted Dobson, Mikhail Muzychuk, Pablo Spiga . . . . . . . . . . . . . . . . 287
The antiprism of an abstract polytope
Ian Gleason, Isabel Hubard . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Linkedness of Cartesian products of complete graphs
Leif K. Jørgensen, Guillermo Pineda-Villavicencio, Julien Ugon . . . . . . 317
Characterization of a family of rotationally symmetric spherical
quadrangulations
Lowell Abrams, Daniel Slilaty . . . . . . . . . . . . . . . . . . . . . . . . 327
Volume 22, Number 2, Spring/Summer 2022, Pages 163–361
xi

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.01 / 163–205
https://doi.org/10.26493/1855-3974.2352.eaf
(Also available at http://amc-journal.eu)
Tight relative t-designs on two shells in
hypercubes, and Hahn and Hermite
polynomials*
Eiichi Bannai
Faculty of Mathematics, Kyushu University (emeritus), Japan, and
National Center for Theoretical Sciences, National Taiwan University, Taiwan
Etsuko Bannai
National Center for Theoretical Sciences, National Taiwan University, Taiwan
Hajime Tanaka †
Research Center for Pure and Applied Mathematics, Graduate School of Information
Sciences, Tohoku University, Sendai 980-8579, Japan
Yan Zhu ‡
College of Science, University of Shanghai for Science and Technology,
Shanghai 200093, China
Received 3 June 2020, accepted 24 May 2021, published online 14 April 2022
Abstract
Relative t-designs in the n-dimensional hypercubeQn are equivalent to weighted regu-
lar t-wise balanced designs, which generalize combinatorial t-(n, k, λ) designs by allowing
multiple block sizes as well as weights. Partly motivated by the recent study on tight Eu-
clidean t-designs on two concentric spheres, in this paper we discuss tight relative t-designs
inQn supported on two shells. We show under a mild condition that such a relative t-design
induces the structure of a coherent configuration with two fibers. Moreover, from this struc-
ture we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal
polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool
to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn
polynomials when they degenerate to the Hermite polynomials under an appropriate limit
*This work was also partially supported by the Research Institute for Mathematical Sciences at Kyoto Univer-
sity.
†Corresponding author. Hajime Tanaka was supported by JSPS KAKENHI Grant Numbers JP25400034 and
JP17K05156.
‡Yan Zhu was supported by NSFC Grant No. 11801353.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
164 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
process, we prove a theorem which gives a partial evidence that the non-trivial tight relative
t-designs in Qn supported on two shells are rare for large t.
Keywords: Relative t-design, association scheme, coherent configuration, Terwilliger algebra, Hahn
polynomial, Hermite polynomial.
Math. Subj. Class. (2020): 05B30, 05E30, 33C45
1 Introduction
This paper is a contribution to the study of relative t-designs in Q-polynomial associa-
tion schemes. In the Delsarte theory [16], the concept of t-designs is introduced for arbi-
trary Q-polynomial association schemes. For the Johnson scheme J(n, k), the t-designs
in the sense of Delsarte are shown to be the same thing as the combinatorial t-(n, k, λ)
designs. There are similar interpretations of t-designs in some other important families of
Q-polynomial association schemes [16, 17, 19, 34, 41]. The concept of relative t-designs
is also due to Delsarte [18], and is a relaxation of that of t-designs. Relative t-designs can
again be interpreted in several cases, including J(n, k). For the n-dimensional hypercube
Qn (or the binary Hamming scheme H(n, 2)) which will be our central focus in this paper,
these are equivalent to the weighted regular t-wise balanced designs, which generalize the
combinatorial t-(n, k, λ) designs by allowing multiple block sizes as well as weights.
The Delsarte theory has a counterpart for the unit sphere Sn−1 in Rn, established
by Delsarte, Goethals, and Seidel [20]. The t-designs in Sn−1 are commonly called the
spherical t-designs, and are essentially the equally-weighted cubature formulas of degree
t for the spherical integration, a concept studied extensively in numerical analysis. Spher-
ical t-designs were later generalized to Euclidean t-designs by Neumaier and Seidel [35]
(cf. [21]). Euclidean t-designs are in general supported on multiple concentric spheres in
Rn, and it follows that we may think of them as the natural counterpart of relative t-designs
in Rn. This point of view was discussed in detail by Bannai and Bannai [3]. See also [7, 8].
The success and the depth of the theory of Euclidean t-designs (cf. [38]) has been one driv-
ing force for the recent research activity on relative t-designs in Q-polynomial association
schemes; see, e.g., [3, 5, 6, 7, 8, 9, 11, 32, 51, 53, 54].
A relative t-design in a Q-polynomial association scheme (X,R) is often defined as
a certain weighted subset of the vertex set X , i.e., a pair (Y, ω) of a subset Y of X and
a function ω : Y → (0,∞). We are given in advance a ‘base vertex’ x ∈ X , and (Y, ω)
gives a ‘degree-t approximation’ of the shells (or spheres or subconstituents) with respect
to x on which Y is supported. See Sections 2 and 3 for formal definitions. Bannai and
Bannai [3] proved a Fisher-type lower bound on |Y |, and we call (Y, ω) tight if it attains
this bound. We may remark that t must be even in this case. In this paper, we continue the
study (cf. [5, 9, 32, 51, 53]) of tight relative t-designs in the hypercubesQn, which are one
of the most important families of Q-polynomial association schemes. The Delsarte theory
directly applies to the tight relative t-designs in Qn supported on one shell, say, the kth
shell, as these are equivalent to the tight combinatorial t-(n, k, λ) designs. (We note that
the kth shell induces J(n, k).) Our aim is to extend this structure theory to those supported
on two shells. We may view the results of this paper roughly as counterparts to (part of) the
E-mail addresses: bannai@math.kyushu-u.ac.jp (Eiichi Bannai), et-ban@rc4.so-net.ne.jp (Etsuko Bannai),
htanaka@tohoku.ac.jp (Hajime Tanaka), zhuyan@usst.edu.cn (Yan Zhu)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 165
results by Bannai and Bannai [2, 4] on tight Euclidean t-designs on two concentric spheres.
Let t = 2e be even. In Theorem 5.3, which is our first main result, we show under
a mild condition that a tight relative 2e-design in Qn supported on two shells induces the
structure of a coherent configuration with two fibers. Moreover, from this structure we de-
duce that a certain polynomial of degree e, known as a Hahn polynomial, must have only
integral simple zeros. We note that the case e = 1 was handled previously by Bannai,
Bannai, and Bannai [5]. The Hahn polynomials are a family of hypergeometric orthogonal
polynomials in the Askey scheme [31, Section 1.5], and that their zeros are integral pro-
vides quite a strong necessary condition on the existence of such relative 2e-designs. The
corresponding necessary condition for the tight combinatorial 2e-(n, k, λ) designs from
the Delsarte theory was used successfully by Bannai [1]; that is to say, he showed that, for
each given integer e ⩾ 5, there exist only finitely many non-trivial tight 2e-(n, k, λ) de-
signs, where n and k (and thus λ) vary. See also [22, 36, 52]. We extend Bannai’s method
to prove our second main result, Theorem 7.1, which presents a version of his theorem for
our case.
The sections other than Sections 5 and 7 are organized as follows. We collect the
necessary background material in Sections 2 and 3. Section 3 also includes a few general
results on relative t-designs in Q-polynomial association schemes. As in [6, 44], our main
tool in the analysis of relative t-designs is the Terwilliger algebra [46, 47, 48], which is
a non-commutative semisimple C-algebra containing the adjacency algebra. Section 4 is
devoted to detailed descriptions of the Terwilliger algebra ofQn. It is well known (cf. [30,
31]) that the Hahn polynomials (3F2) degenerate to the Hermite polynomials (2F0) by an
appropriate limit process, and a key in Bannai’s method above was to evaluate precisely the
behavior of the zeros of the Hahn polynomials in this process. In Section 6, we revisit this
part of the method in a form suited to our purpose. Our account will also be simpler than
that in [1]. In Appendix, we provide a proof of a number-theoretic result (Proposition 7.2)
which is a variation of a result of Schur [40, Satz I].
2 Coherent configurations and association schemes
We begin by recalling the concept of coherent configurations.
Definition 2.1. The pair (X,R) of a finite set X and a set R of non-empty subsets of X2
is called a coherent configuration on X if it satisfies the following (C1) – (C4):
(C1) R is a partition of X2.
(C2) There is a subset R0 of R such that⊔
R∈R0
R = {(x, x) : x ∈ X}.
(C3) R is invariant under the transposition τ : (x, y) 7→ (y, x) ((x, y) ∈ X2), i.e.,
Rτ ∈ R for all R ∈ R.
(C4) For all R,S, T ∈ R and (x, y) ∈ T , the number
pTR,S :=
∣∣{z ∈ X : (x, z) ∈ R, (z, y) ∈ S}∣∣
is independent of the choice of (x, y) ∈ T .
166 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
Moreover, a coherent configuration (X,R) on X is called homogeneous if |R0| = 1, and
an association scheme if Rτ = R for all R ∈ R.
Remark 2.2. Suppose that a finite group G acts on X , and let R be the set of the orbitals
of G, that is to say, the orbits of G in its natural action on X2. Then (X,R) is a coherent
configuration. Moreover, (X,R) is homogeneous (resp. an association scheme) if and only
if the action of G on X is transitive (resp. generously transitive, i.e., for any x, y ∈ X we
have (xg, yg) = (y, x) for some g ∈ G).
Let (X,R) be a coherent configuration as above. For every R ∈ R0, let ΦR be the
subset of X such that R = {(x, x) : x ∈ ΦR}. Then we have⊔
R∈R0
ΦR = X.
We call the ΦR (R ∈ R0) the fibers of (X,R). By setting in (C4) either R ∈ R0 and
S = T , or S ∈ R0 and R = T , it follows that for every T ∈ R, we have T ⊂ ΦR × ΦS
for some R,S ∈ R0. In particular, (X,R) is homogeneous whenever it is an association
scheme. Let
γR,S =
∣∣{T ∈ R : T ⊂ ΦR × ΦS}∣∣ (R,S ∈ R0).
The matrix
[γR,S ]R,S∈R0 ,
which is symmetric by (C3), is called the type of (X,R).
Let MX(C) be the C-algebra of all complex matrices with rows and columns indexed
by X , and let V = CX be the C-vector space of complex column vectors with coordinates
indexed by X . We endow V with the Hermitian inner product
⟨u, v⟩ = v†u (u, v ∈ V ),
where † denotes adjoint. For every R ∈ R, let AR ∈ MX(C) be the adjacency matrix of
the graph (X,R) (directed, in general), i.e.,
(AR)x,y =
{
1 if (x, y) ∈ R,
0 otherwise,
(x, y ∈ X).
Then (C1) – (C4) above are rephrased as follows:
(A1)
∑
R∈R
AR = J (the all-ones matrix).
(A2)
∑
R∈R0
AR = I (the identity matrix).
(A3) (AR)† ∈ {AS : S ∈ R} (R ∈ R).
(A4) ARAS =
∑
T∈R
pTR,SAT (R,S ∈ R).
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 167
Let
A = span{AR : R ∈ R}.
Then from (A2) and (A4) it follows that A is a subalgebra ofMX(C), called the adjacency
algebra of (X,R). We note that A is semisimple as it is closed under † by virtue of (A3).
By (A1), A is also closed under entrywise (or Hadamard or Schur) multiplication, which
we denote by ◦. The AR are the (central) primitive idempotents of A with respect to ◦, i.e.,
AR ◦AS = δR,SAR,
∑
R∈R
AR = J.
Remark 2.3. If (X,R) arises from a group action as in Remark 2.2, then A coincides with
the centralizer algebra (or Hecke algebra or commutant) for the corresponding permutation
representation g 7→ Pg (g ∈ G) on V , i.e.,
A = {B ∈MX(C) : BPg = PgB (g ∈ G)}.
A subalgebra of MX(C) is called a coherent algebra if it contains J , and is closed
under ◦ and †. We note that the coherent algebras are precisely the adjacency algebras of
coherent configurations. It is clear that the intersection of coherent algebras in MX(C) is
again a coherent algebra. In particular, for any subset S of MX(C), we can speak of the
smallest coherent algebra containing S, which we call the coherent closure of S.
From now on, we assume that (X,R) is an association scheme. As is the case for many
examples of association schemes, we write
R = {R0, R1, . . . , Rn}, where R0 = {R0},
and say that (X,R) has n classes. We will then abbreviate pki,j = p
Rk
Ri,Rj
, Ai = ARi ,
and so on. The adjacency algebra A is commutative in this case, and hence it has a basis
E0, E1, . . . , En consisting of the (central) primitive idempotents, i.e.,
EiEj = δi,jEi,
n∑
i=0
Ei = I.
Put differently, E0V,E1V, . . . , EnV are the maximal common eigenspaces (or homoge-
neous components or isotypic components) of A, and the Ei are the corresponding orthog-
onal projections. Since the Ai are real symmetric matrices, so are the Ei. Note that the
matrix |X|−1J ∈ A is an idempotent with rank one, and thus primitive. We will always
set
E0 =
1
|X|
J.
For convenience, we let
Ai = Ei := O (the zero matrix) if i < 0 or i > n.
Though our focus in this paper will be on Q-polynomial association schemes, we first
recall the P -polynomial property for completeness. We say that the association scheme
(X,R) is P -polynomial (or metric) with respect to the ordering A0, A1, . . . , An if there
168 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
are non-negative integers ai, bi, ci (0 ⩽ i ⩽ n) such that bn = c0 = 0, bi−1ci ̸= 0
(1 ⩽ i ⩽ n), and
A1Ai = bi−1Ai−1 + aiAi + ci+1Ai+1 (0 ⩽ i ⩽ n),
where b−1 and cn+1 are indeterminates. In this case, A1 recursively generates A, and
hence has n+ 1 distinct eigenvalues θ0, θ1, . . . , θn ∈ R, where we write
A1 =
n∑
i=0
θiEi. (2.1)
We note that (X,R) is P -polynomial as above precisely when the graph (X,R1) is a
distance-regular graph and (X,Ri) is the distance-i graph of (X,R1) (0 ⩽ i ⩽ n). See,
e.g., [10, 12, 27, 15] for more information on distance-regular graphs.
We say that (X,R) is Q-polynomial (or cometric) with respect to the ordering
E0, E1, . . . , En if there are real scalars a∗i , b
∗
i , c
∗
i (0 ⩽ i ⩽ n) such that b
∗
n = c
∗
0 = 0,
b∗i−1c
∗
i ̸= 0 (1 ⩽ i ⩽ n), and
E1 ◦ Ei =
1
|X|
(b∗i−1Ei−1 + a
∗
iEi + c
∗
i+1Ei+1) (0 ⩽ i ⩽ n), (2.2)
where b∗−1 and c
∗
n+1 are indeterminates. In this case, |X|E1 recursively generates A with
respect to ◦, and hence has n+ 1 distinct entries θ∗0 , θ∗1 , . . . , θ∗n ∈ R, where we write
|X|E1 =
n∑
i=0
θ∗iAi. (2.3)
We call the θ∗i the dual eigenvalues of |X|E1. We may remark that E1 ◦ Ei, being a
principal submatrix of E1 ⊗ Ei, is positive semidefinite, so that the scalars a∗i , b∗i , and c∗i
are non-negative (the so-called Krein condition). The Q-polynomial association schemes
are an important subject in their own right, and we refer the reader to [23, 29] and the
references therein for recent activity.
Below we give two fundamental examples ofP - andQ-polynomial association schemes,
both of which come from transitive group actions. See [10, 12, 16] for the details.
Example 2.4. Let v and k be positive integers with v > k, and letX be the set of k-subsets
of {1, 2, . . . , v}. Set n = min{k, v − k}. For x, y ∈ X and 0 ⩽ i ⩽ n, we let (x, y) ∈ Ri
if |x ∩ y| = k − i. The Ri are the orbitals of the symmetric group Sv acting on X . We
call (X,R) a Johnson scheme and denote it by J(v, k). The eigenvalues of A1 are given
in decreasing order by
θi = (k − i)(v − k − i)− i (0 ⩽ i ⩽ n),
and J(v, k) isQ-polynomial with respect to the corresponding ordering of theEi (cf. (2.1)).
Example 2.5. Let q ⩾ 2 be an integer and let X = {0, 1, . . . , q − 1}n. For x, y ∈ X and
0 ⩽ i ⩽ n, we let (x, y) ∈ Ri if x and y differ in exactly i coordinate positions. The Ri
are the orbitals of the wreath product Sq ≀Sn of the symmetric groups Sq and Sn acting
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 169
on X . We call (X,R) a Hamming scheme and denote it by H(n, q). The eigenvalues of
A1 are given in decreasing order by
θi = n(q − 1)− qi (0 ⩽ i ⩽ n),
andH(n, q) isQ-polynomial with respect to the corresponding ordering of theEi (cf. (2.1)).
The Hamming scheme H(n, 2) is also known as the n-cube (or n-dimensional hypercube)
and is denoted by Qn.
Assumption 2.6. For the rest of this section and in Section 3, we assume that (X,R) is
an association scheme and is Q-polynomial with respect to the ordering E0, E1, . . . , En of
the primitive idempotents.
In general, for any positive semidefinite Hermitian matrices B,C ∈ MX(C), we have
(cf. [45])
(B ◦ C)V = span(BV ◦ CV ),
where
BV ◦ CV = {u ◦ v : u ∈ BV, v ∈ CV }.
Hence it follows from (2.2) that
span(E1V ◦ EiV ) =
{
Ei−1V + EiV + Ei+1V if a∗i ̸= 0,
Ei−1V + Ei+1V if a∗i = 0,
(0 ⩽ i ⩽ n), (2.4)
from which it follows that
h∑
i=0
k∑
j=0
span(EiV ◦ EjV ) =
h∑
i=0
k∑
j=0
span(E1V ◦ · · · ◦ E1V︸ ︷︷ ︸
i times
◦EjV )
=
h+k∑
i=0
EiV (2.5)
for 0 ⩽ h, k ⩽ n. See also [10, Section 2.8].
We now fix a ‘base vertex’ x ∈ X . Let
Xi = {y ∈ X : (x, y) ∈ Ri} (0 ⩽ i ⩽ n).
We call the Xi the shells (or spheres or subconstituents) of (X,R) with respect to x. For
every i (0 ⩽ i ⩽ n), define the diagonal matrix E∗i = E
∗
i (x) ∈MX(C) by
(E∗i )y,y =
{
1 if y ∈ Xi,
0 otherwise,
(y ∈ X).
Then we have
E∗i E
∗
j = δi,jE
∗
i ,
n∑
i=0
E∗i = I.
We call the E∗i the dual idempotents of (X,R) with respect to x. The subspace
A∗ = A∗(x) = span{E∗0 , E∗1 , . . . , E∗n}
170 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
is then a subalgebra of MX(C), which we call the dual adjacency algebra of (X,R) with
respect to x. The Terwilliger algebra (or subconstituent algebra) of (X,R) with respect to
x is the subalgebra T = T (x) of MX(C) generated by A and A∗ [46, 47, 48]. We note
that T is semisimple as it is closed under †.
Remark 2.7. If (X,R) arises from a group action as in Remark 2.2, which we recall is
generously transitive in this case, then T is a subalgebra of the centralizer algebra for the
action of the stabilizer Gx of x in G. The two algebras are known to be equal, e.g., for
J(v, k) and H(n, q); see [25, 43].
For every subset Y of X , let Ŷ ∈ V be the characteristic vector of Y , i.e.,
(Ŷ )y =
{
1 if y ∈ Y,
0 otherwise,
(y ∈ X).
In particular, X̂ denotes the all-ones vector in V . We will simply write x̂ for the character-
istic vector of the singleton {x}. With this notation established, we have
X̂i = E
∗
i X̂ = Aix̂ (0 ⩽ i ⩽ n),
from which it follows that
T x̂ = span{X̂i : 0 ⩽ i ⩽ n} = span{Eix̂ : 0 ⩽ i ⩽ n}. (2.6)
The T -module T x̂ is easily seen to be irreducible with dimension n + 1 (cf. [46, Lem-
ma 3.6]), and is called the primary T -module.
We define the dual adjacency matrix A∗1 = A
∗
1(x) ∈MX(C) by (cf. (2.3))
A∗1 = |X|diagE1x̂ =
n∑
i=0
θ∗iE
∗
i . (2.7)
Since the θ∗i are mutually distinct, A
∗
1 generates A
∗. Moreover, since
A∗1v = |X|(E1x̂) ◦ v (v ∈ V ),
it follows from (2.4) that
EiA
∗
1Ej = O if |i− j| > 1 (0 ⩽ i, j ⩽ n). (2.8)
Let W be an irreducible T -module. We define the dual support W ∗s , the dual endpoint
r∗(W ), and the dual diameter d∗(W ) of W by
W ∗s = {i : EiW ̸= 0}, r∗(W ) = minW ∗s , d∗(W ) = |W ∗s | − 1,
respectively. We call W dual thin if dimEiW ⩽ 1 (0 ⩽ i ⩽ n). We note that the
primary T -module T x̂ is dual thin, and that it is a unique irreducible T -module up to
isomorphism which has dual endpoint zero or dual diameter n. The following lemma is an
easy consequence of (2.8):
Lemma 2.8 ([46, Lemma 3.12]). With reference to Assumption 2.6, write A∗1 = A∗1(x),
A∗ = A∗(x), T = T (x). Let W be an irreducible T -module and set r∗ = r∗(W ),
d∗ = d∗(W ). Then the following hold:
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 171
1. A∗1EiW ⊂ Ei−1W + EiW + Ei+1W (0 ⩽ i ⩽ n).
2. W ∗s = {r∗, r∗ + 1, . . . , r∗ + d∗}.
3. EiA∗1EjW ̸= 0 if |i− j| = 1 (r∗ ⩽ i, j ⩽ r∗ + d∗).
4. Suppose that W is dual thin. Then
i∑
h=0
Er∗+hW =
i∑
h=0
(A∗1)
hEr∗W (0 ⩽ i ⩽ d
∗).
In particular, W = A∗Er∗W .
3 Relative t-designs in Q-polynomial association schemes
In this section, we develop some general theory on relative t-designs in Q-polynomial
association schemes.
Recall Assumption 2.6. Throughout this section, we fix a base vertex x ∈ X , and write
E∗i = E
∗
i (x) (0 ⩽ i ⩽ n), A
∗
1 = A
∗
1(x), A
∗ = A∗(x), and T = T (x). In Introduction,
we meant by a weighted subset of X a pair (Y, ω) of a subset Y of X and a function
ω : Y → (0,∞). For convenience, however, we extend the domain of ω to X by setting
ω(y) = 0 for every y ∈ X\Y . We will also naturally identify V with the set of complex
functions on X , so that ω ∈ V and Y = suppω. In our discussions on relative t-designs,
we will often consider the set
L = LY = {ℓ : Y ∩Xℓ ̸= ∅}, (3.1)
and say that (Y, ω) is supported on
⊔
ℓ∈LXℓ.
For comparison, we begin with the algebraic definition of t-designs in (X,R) due to
Delsarte [16, 17].
Definition 3.1. A weighted subset (Y, ω) of X is called a t-design in (X,R) if Eiω = 0
for 1 ⩽ i ⩽ t.
Delsarte [18] generalized this concept as follows:
Definition 3.2. A weighted subset (Y, ω) of X is called a relative t-design in (X,R) (with
respect to x) if Eiω ∈ span{Eix̂} for 1 ⩽ i ⩽ t.
Remark 3.3. Delsarte introduced the concept of t-designs for subsets Y of X in [16], i.e.,
when ω = Ŷ , whereas in [17, 18] he mostly considered general (i.e., not necessarily non-
negative) non-zero vectors ω ∈ V in the discussions on t-designs and relative t-designs.
Some facts/results below, such as Examples 3.4 and 3.5, Proposition 3.6, and Theorem 3.8,
are still valid for general ω ∈ V , but the Fisher-type lower bound on |Y | = | suppω|
(cf. Theorem 3.9) makes sense only when ω is non-negative.
For the Johnson and Hamming schemes, Delsarte [16, 17, 18] showed that these alge-
braic concepts indeed have geometric interpretations:
Example 3.4. Let (X,R) be the Johnson scheme J(v, k) from Example 2.4. Then (Y, ω)
is a t-design if and only if, for every t-subset z of {1, 2, . . . , v}, the sum λz of the values
172 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
ω(y) over those y ∈ Y such that z ⊂ y, is a constant independent of z. On the other hand,
(Y, ω) is a relative t-design if and only if the above λz depends only on |x ∩ z|. We note
that (Y, Ŷ ) is a t-design if and only if Y is a t-(v, k, λ) design (cf. [13, Chapter II.4]) for
some λ.
Example 3.5. Let (X,R) be the Hamming scheme H(n, q) from Example 2.5. Then
(Y, ω) is a t-design if and only if, for every t-subset T of {1, 2, . . . , n} and every function
f : T → {0, 1, . . . , q − 1}, the sum λT ,f of the values ω(y) over those
y = (y1, y2, . . . , yn) ∈ Y such that yi = f(i) (i ∈ T ), is a constant independent of
the pair (T , f). On the other hand, (Y, ω) is a relative t-design if and only if the above
λT ,f depends only on |{i ∈ T : xi = f(i)}|, where x = (x1, x2, . . . , xn). We note that
(Y, Ŷ ) is a t-design if and only if the transpose of the |Y | × n matrix formed by arranging
the elements of Y (in any order) is an orthogonal array OA(|Y |, n, q, t) (cf. [13, Chapter
III.6]). For the case q = 2, i.e., for Qn, if we choose the base vertex as x = (0, 0, . . . , 0),
then (Y, Ŷ ) is a relative t-design if and only if Y is a regular t-wise balanced design of type
t-(n,L, λ) (cf. [38, Section 4.4]) for some λ, where L is from (3.1), and where we identify
the elements of X = {0, 1}n with their supports.
Similar results hold for some other important families of P - and Q-polynomial association
schemes; see, e.g., [17, 18, 19, 34, 41].
Proposition 3.6 (cf. [3, Theorem 4.5]). With reference to Assumption 2.6, let (Y, ω) be a
weighted subset supported on
⊔
ℓ∈LXℓ. Then we have
ω|T x̂ =
∑
ℓ∈L
⟨ω, X̂ℓ⟩
|Xℓ|
X̂ℓ, (3.2)
where ω|T x̂ denotes the orthogonal projection of ω on the primary T -module T x̂. More-
over, (Y, ω) is a relative t-design if and only if
⟨ω, v⟩ = ⟨ω|T x̂, v⟩ =
∑
ℓ∈L
⟨ω, X̂ℓ⟩
|Xℓ|
⟨X̂ℓ, v⟩
for every v ∈
∑t
i=0EiV .
Proof. Recall (2.6). The first part follows since the X̂i form an orthogonal basis of T x̂
with ∥X̂i∥2 = |Xi|. The second part is also immediate from
Eiω ∈ span{Eix̂} ⇐⇒ Eiω ∈ T x̂ ⇐⇒ Eiω|T x̂ = Eiω.
Remark 3.7. It is clear that (Xℓ, X̂ℓ) is a relative n-design for every 0 ⩽ ℓ ⩽ n. Hence,
if (Y, ω) is a relative t-design such that Xℓ ⊂ Y for some ℓ, and if ω is constant on Xℓ,
then the weighted subset (Y \Xℓ, (I − E∗ℓ )ω) obtained by discarding Xℓ from Y is again
a relative t-design. This observation is particularly important when applying Theorem 3.8
below; for example, we can always assume that 0 ̸∈ L.
The following is a slight generalization of Delsarte’s Assmus–Mattson theorem for Q-
polynomial association schemes [18, Theorem 8.4], and can also be viewed as a variation
of [9, Theorem 3.3], which in turn generalizes [28, Proposition 1]. See also [11]. The
proof is in fact identical to that of [44, Theorem 4.3], but we include it below because of
the potential importance of the result.
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 173
Theorem 3.8. With reference to Assumption 2.6, let (Y, ω) be a relative t-design supported
on
⊔
ℓ∈LXℓ. Then (Y ∩Xℓ, E∗ℓω) is a relative (t− |L|+ 1)-design for every ℓ ∈ L.
Proof. Let U = (T x̂)⊥ be the orthogonal complement of T x̂ in V , which we recall is the
sum of all the non-primary irreducible T -modules in V . On the one hand, we have
ω|U ∈
∑
ℓ∈L
E∗ℓU.
Since A∗1 generates A
∗ and has at most |L| distinct eigenvalues on this subspace (cf. (2.7)),
it follows that
A∗ω|U = span
{
ω|U , A∗1ω|U , . . . , (A∗1)|L|−1ω|U
}
. (3.3)
On the other hand, since E0U = 0, that (Y, ω) is a relative t-design is rephrased as
ω|U ∈
n∑
i=t+1
EiU.
Hence it follows from (2.8) and (3.3) that
A∗ω|U ⊂
|L|−1∑
k=0
(A∗1)
k
n∑
i=t+1
EiU ⊂
n∑
i=t−|L|+2
EiU.
In particular, for every ℓ ∈ L we have
E∗ℓω|U ∈
n∑
i=t−|L|+2
EiU.
In other words, (Y ∩Xℓ, E∗ℓω) is a relative (t− |L|+ 1)-design, as desired.
Bannai and Bannai [3, Theorem 4.8] established the following Fisher-type lower bound
on the size of a relative t-design with t even:
Theorem 3.9. With reference to Assumption 2.6, let (Y, ω) be a relative 2e-design (e ∈ N)
supported on
⊔
ℓ∈LXℓ. Then
|Y | ⩾ dim
(∑
ℓ∈L
E∗ℓ
)(
e∑
i=0
EiV
)
.
Definition 3.10. A relative 2e-design (Y, ω) is called tight if equality holds above.
Recall from Example 3.5 that the relative t-designs in the hypercubes are equivalent to
the weighted regular t-wise balanced designs.
Example 3.11. Let (X,R) be the n-cube Qn from Example 2.5. Xiang [51] showed that
if e ⩽ ℓ ⩽ n− e for every ℓ ∈ L, then
dim
(∑
ℓ∈L
E∗ℓ
)(
e∑
i=0
EiV
)
=
min{|L|−1,e}∑
i=0
(
n
e− i
)
. (3.4)
174 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
We may remark that (cf. [12, Theorem 9.2.1])
dimEiV =
(
n
i
)
(0 ⩽ i ⩽ n). (3.5)
See also [32] and [6, Theorem 2.7, Example 2.9].
Example 3.12. Consider a symmetric 2-(n + 1, k, λ) design (cf. [13, Chapter II.6]). Ob-
serve that removing a point yields a tight relative 2-design in Qn with L = {k − 1, k}.
Likewise, taking the complement of every block which contains a given point followed by
removing that point gives rise to a tight relative 2-design in Qn with L = {k, n+ 1− k}.
The complement of this is yet another example1 such that L = {k − 1, n − k}. See [32,
Section 3] and [50, Theorem 8]. Note that the weights are constant for these three exam-
ples. On the other hand, Bannai, Bannai, and Bannai [5, Theorem 2.2] showed that there
is a tight relative 2-design in Qn with L = {2, n/2} for n ≡ 6 (mod 8), provided that a
Hadamard matrix of order n/2 + 1 exists. This construction provides examples in which
the weights take two distinct values depending on the shells. See also [53].
Example 3.13. Working with the tight 4-(23, 7, 1) and 4-(23, 16, 52) designs instead of a
symmetric 2-(n+1, k, λ) design as in Example 3.12, we obtain four tight relative 4-designs
in Q22 with constant weight such that
L ∈
{
{6, 7}, {6, 15}, {7, 16}, {15, 16}
}
.
See [9, Theorem 6.3] and [32, Section 3].
Let (Y, ω) be a tight relative 2e-design supported on
⊔
ℓ∈LXℓ. Bannai, Bannai, and
Bannai [5, Theorem 2.1] showed that if the stabilizer of x in the automorphism group of
(X,R) acts transitively on each of the shells Xi then ω is constant on Y ∩ Xℓ for every
ℓ ∈ L. The next theorem generalizes this result by replacing group actions by combinatorial
regularity. Observe that the fibers of the coherent closure of T are in general finer than the
shells Xi.
Theorem 3.14. With reference to Assumption 2.6, let (Y, ω) be a tight relative 2e-design
(e ∈ N) supported on
⊔
ℓ∈LXℓ. For every ℓ ∈ L, the weight ω is constant on Y ∩ Xℓ
provided that Xℓ remains a fiber of the coherent closure of T .
Proof. Let (cf. (3.2))
D = diagω, D̃ = diagω|T x̂ =
∑
ℓ∈L
⟨ω, X̂ℓ⟩
|X̂ℓ|
E∗ℓ .
Note that D̃ ∈ T . Let F be the orthogonal projection onto BV , where
B =
√
D̃
e∑
i=0
Ei ∈ T .
Observe that
BV = (BB†)V,
1It seems that this construction is missing in [50, Theorem 8].
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 175
and that F is written as a polynomial in the Hermitian (in fact, real and symmetric) matrix
BB†. In particular, F ∈ T .
Since (Y, ω) is tight, we have
dimBV = dim
√
D̃
(∑
ℓ∈L
E∗ℓ
)(
e∑
i=0
EiV
)
= |Y |.
Let u1, u2, . . . , u|Y | be an orthonormal basis of BV , and let
G =
[
u1 u2 · · · u|Y |
]
.
Then we have
F = GG†. (3.6)
Let
D′ = D|Y×Y , D̃′ = D̃|Y×Y , F ′ = F |Y×Y , G′ = G|Y×{1,2,...,|Y |},
where |Y×Y etc. mean taking corresponding submatrices. Note that these are square matri-
ces, and that D′ and D̃′ are invertible. Then it follows that
(G′)†D′(D̃′)−1G′ = I|Y |. (3.7)
Indeed, since we may write
ui =
√
D̃ vi, where vi ∈
e∑
r=0
ErV (1 ⩽ i ⩽ |Y |),
it follows from (2.5) (applied to h = k = e) and Proposition 3.6 that the (i, j)-entry of the
LHS in (3.7) is equal to
(vi)
†Dvj = ⟨ω, vi ◦ vj⟩ = ⟨ω|T x̂, vi ◦ vj⟩ = (vi)†D̃vj = ⟨uj , ui⟩ = δi,j ,
where means complex conjugate. By (3.6) and (3.7), we have
I|Y | = D
′(D̃′)−1G′(G′)† = D′(D̃′)−1F ′,
so that
(D′)−1 = (D̃′)−1F ′. (3.8)
In particular, F ′ is a diagonal matrix.
Now, let ℓ ∈ L and suppose that Xℓ remains a fiber of the coherent closure of T . Then
the (y, y)-entry of F ∈ T is constant for y ∈ Xℓ (cf. (A1) and (A2)), and the same is true
for D̃. Hence it follows from (3.8) that ω(y) = Dy,y must be constant for y ∈ Y ∩ Xℓ.
This completes the proof.
176 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
4 The Terwilliger algebra of Qn
For the rest of this paper, we will focus on relative t-designs in the n-cube Qn from Ex-
ample 2.5. We will need detailed descriptions of the Terwilliger algebra ofQn and its irre-
ducible modules, and we collect these in this section. Thus, we assume that (X,R) = Qn,
where X = {0, 1}n. We again fix a base vertex x ∈ X , and write E∗i = E∗i (x)
(0 ⩽ i ⩽ n), A∗1 = A
∗
1(x), and T = T (x). The Q-polynomial ordering we consider
is the one given in Example 2.5.2
Proposition 4.1 (cf. [39, Section I.C]). We have
T = span{E∗i AjE∗k : 0 ⩽ i, j, k ⩽ n}. (4.1)
In particular, T is a coherent algebra.
Proof. The RHS in (4.1) is a subspace of T . Recall from Example 2.5 that Qn admits
the action of G = S2 ≀ Sn. The stabilizer Gx of x in G is isomorphic to Sn, and it is
immediate to see that every orbital of Gx is of the form
{(y, z) ∈ X ×X : (x, y) ∈ Ri, (y, z) ∈ Rj , (z, x) ∈ Rk}
for some i, j, and k, where the corresponding adjacency matrix is E∗i AjE
∗
k . Hence the
RHS in (4.1) agrees with the centralizer algebra for the action of Gx on X , which is a
coherent algebra; cf. Remark 2.3. Since T is generated by the Ai and the E∗i , the result
follows.
Lemma 4.2. For 0 ⩽ i, j, k ⩽ n, we have E∗i AjE∗k ̸= O if and only if
j ∈
{
|i− k|, |i− k|+ 2, |i− k|+ 4, . . . ,min{i+ k, 2n− i− k}
}
.
Proof. Routine.
Next we recall basic facts about the irreducible T -modules. Let W be an irreducible
T -module. We define the support Ws, the endpoint r(W ), and the diameter d(W ) of W
by
Ws = {i : E∗iW ̸= 0}, r(W ) = minWs, d(W ) = |Ws| − 1,
respectively. We call W thin if dimE∗iW ⩽ 1 (0 ⩽ i ⩽ n).
Theorem 4.3 (cf. [26]). Let W be an irreducible T -module and set r = r(W ),
r∗ = r∗(W ), d = d(W ), and d∗ = d∗(W ). Then W is thin, dual thin, and we have
r = r∗, d = d∗ = n− 2r, Ws =W ∗s = {r, r + 1, . . . , n− r}.
Moreover, the isomorphism class of W is determined by r.
Remark 4.4. Recall that the universal enveloping algebra U(sl2(C)) is defined by the
generators x, y, h and the relations
xy− yx = h, hx− xh = 2x, hy− yh = −2y.
2If n is even then Qn has another Q-polynomial ordering E0, En−1, E2, En−3, . . . in terms of the natural
ordering; cf. [10, p. 305].
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 177
There is a surjective homomorphism U(sl2(C))→ T such that (cf. [26, Lemma 7.5])
x 7→
n∑
i=1
Ei−1A
∗
1Ei, y 7→
n−1∑
i=0
Ei+1A
∗
1Ei, h 7→ A1.
Every irreducible T -module is then irreducible as an sl2(C)-module. We also obtain an-
other surjective homomorphism U(sl2(C)) → T by interchanging A1 and A∗1 and replac-
ing the Ei by the E∗i above; cf. [26, Lemma 5.3].
From now on, we fix an orthogonal irreducible decomposition
V =
⊕
W∈Λ
W (4.2)
of the standard module V . In view of Theorem 4.3, let
Λr = {W ∈ Λ : r(W ) = r∗(W ) = r} (0 ⩽ r ⩽ ⌊n/2⌋), (4.3)
and fix a unit vector vW ∈ ErW for each W ∈ Λr. Since
dimEiV =
∑
W∈Λ
dimEiW =
i∑
r=0
|Λr| (0 ⩽ i ⩽ ⌊n/2⌋) (4.4)
by Theorem 4.3, it follows from (3.5) that
|Λr| =
(
n
r
)
−
(
n
r − 1
)
(0 ⩽ r ⩽ ⌊n/2⌋).
It is known (cf. [26, Theorem 9.2]) that if W ∈ Λr then the vectors
E∗r vW , E
∗
r+1vW , . . . , E
∗
n−rvW (4.5)
form an orthogonal basis of W , called a standard basis of W . By [26, Lemma 6.6], we
also have
∥E∗i vW ∥2 =
(
n− 2r
i− r
)
∥E∗r vW ∥2 (r ⩽ i ⩽ n− r). (4.6)
We note that
1 = ∥vW ∥2 =
n−r∑
i=r
∥E∗i vW ∥2 = 2n−2r∥E∗r vW ∥2. (4.7)
For W,W ′ ∈ Λr, we observe that the linear map W →W ′ defined by
E∗i vW 7→ E∗i vW ′ (r ⩽ i ⩽ n− r)
is an isometric isomorphism of T -modules. Let
Ĕi,jr =
2n−2r√(
n−2r
i−r
)(
n−2r
j−r
) ∑
W∈Λr
(E∗i vW )(E
∗
j vW )
† (r ⩽ i, j ⩽ n− r). (4.8)
178 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
Then we have
(Ĕi,jr )
† = Ĕj,ir (r ⩽ i, j ⩽ n− r), (4.9)
and from (4.6) and (4.7) it follows that
Ĕi,jr Ĕ
i′,j′
r′ = δr,r′δj,i′Ĕ
i,j′
r
for 0 ⩽ r, r′ ⩽ ⌊n/2⌋, r ⩽ i, j ⩽ n − r, and r′ ⩽ i′, j′ ⩽ n − r′. By Theorem 4.3
and Wedderburn’s theorem (cf. [14, Section 3]), T is isomorphic to the direct sum of full
matrix algebras
T ∼=
⌊n/2⌋⊕
r=0
Mn−2r+1(C),
and the Ĕi,jr form an orthogonal basis of T . See also [24, Section 2]. We note that
E∗i TE
∗
j = span
{
Ĕi,jr : 0 ⩽ r ⩽ min{i, j, n− i, n− j}
}
(0 ⩽ i, j ⩽ n). (4.10)
We now recall the Hahn polynomials [31, Section 1.5]
Qr(ξ;α, β,N) = 3F2
(
−ξ,−r, r + α+ β + 1
α+ 1,−N
∣∣∣∣ 1) ∈ R[ξ] (0 ⩽ r ⩽ N), (4.11)
where
sFt
(
a1, . . . , as
b1, . . . , bt
∣∣∣∣ c) = ∞∑
i=0
(a1)i · · · (as)i
(b1)i · · · (bt)i
ci
i!
,
and
(a)i = a(a+ 1) · · · (a+ i− 1).
For α, β > −1, or for α, β < −N , we have
N∑
ξ=0
(
α+ ξ
ξ
)(
β +N − ξ
N − ξ
)
Qr(ξ;α, β,N)Qr′(ξ;α, β,N)
= δr,r′
(−1)r(r + α+ β + 1)N+1(β + 1)rr!
(2r + α+ β + 1)(α+ 1)r(−N)rN !
. (4.12)
Our aim is to describe the entries of the Ĕi,jr . In view of (4.9), we will assume for the
rest of this section that
0 ⩽ i ⩽ j ⩽ n.
By Proposition 4.1 and Lemma 4.2, we have
E∗i TE
∗
j = span
{
E∗i A2ξ+j−iE
∗
j : 0 ⩽ ξ ⩽ min{i, n− j}
}
.
Moreover, it follows that (cf. (4.10))
E∗i A2ξ+j−iE
∗
j
=
min{i,n−j}∑
r=0
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1)
(
j
i−ξ
)(
n−j
ξ
)(
j−r
j−i
)√(
n−2r
j−r
)
(
j
i
)√(
n−2r
i−r
) Ĕi,jr . (4.13)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 179
This formula can be found in [33, Section 10]. See also [39, 49] for similar calculations.
If i ⩽ n− j then
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1) = Qr(ξ; j − n− 1,−j − 1, i).
Since (
j
i− ξ
)(
n− j
ξ
)
= (−1)i
(
j − n− 1 + ξ
ξ
)(
−j − 1 + i− ξ
i− ξ
)
,
it follows from (4.12) (applied to α = j − n− 1, β = −j − 1, N = i) and (4.13) that, for
0 ⩽ r ⩽ i,
i∑
ξ=0
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1)E∗i A2ξ+j−iE∗j
=
(−1)r(r − n− 1)i+1(−j)rr!
(2r − n− 1)(j − n)r(−i)ri!
·
(−1)i
(
j−r
j−i
)√(
n−2r
j−r
)
(
j
i
)√(
n−2r
i−r
) Ĕi,jr
=
(
n
i
)(
n−i
r
)√(
n−2r
j−r
)((
n
r
)
−
(
n
r−1
))(
n−j
r
)√(
n−2r
i−r
) Ĕi,jr .
Likewise, if n− j ⩽ i then
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1) = Qr(ξ;−i− 1, i− n− 1, n− j).
In this case, since(
j
i− ξ
)(
n− j
ξ
)(
j
i
)−1
= (−1)n−j
(
−i− 1 + ξ
ξ
)(
i− 1− j − ξ
n− j − ξ
)(
n− i
n− j
)−1
,
again it follows from (4.12) (applied to α = −i− 1, β = i−n− 1, N = n− j) and (4.13)
that, for 0 ⩽ r ⩽ n− j,
n−j∑
ξ=0
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1)E∗i A2ξ+j−iE∗j
=
(−1)r(r − n− 1)n−j+1(i− n)rr!
(2r − n− 1)(−i)r(j − n)r(n− j)!
·
(−1)n−j
(
j−r
j−i
)√(
n−2r
j−r
)
(
n−i
n−j
)√(
n−2r
i−r
) Ĕi,jr
=
(
n
i
)(
n−i
r
)√(
n−2r
j−r
)((
n
r
)
−
(
n
r−1
))(
n−j
r
)√(
n−2r
i−r
) Ĕi,jr .
In either case, it follows that
Ĕi,jr =
((
n
r
)
−
(
n
r−1
))(
n−j
r
)√(
n−2r
i−r
)
(
n
i
)(
n−i
r
)√(
n−2r
j−r
)
180 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
×
min{i,n−j}∑
ξ=0
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1)E∗i A2ξ+j−iE∗j (4.14)
for 0 ⩽ i ⩽ j ⩽ n and 0 ⩽ r ⩽ min{i, n− j}.
5 Tight relative 2e-designs on two shells in Qn
We retain the notation of the previous sections. In this section, we discuss tight relative
2e-designs (Y, ω) in Qn supported on two shells Xℓ ⊔ Xm, i.e., L = {ℓ,m} (cf. (3.1)).
Recall from (3.4) that we have in this case
|Y | =
(
n
e
)
+
(
n
e− 1
)
,
but recall also that this is valid under the additional condition that e ⩽ ℓ,m ⩽ n− e. How-
ever, both (Y ∩Xℓ, E∗ℓω) and (Y ∩Xm, E∗mω) are relative (2e− 1)-designs by Theorem
3.8, so that if ℓ < 2e or ℓ > n − 2e for example, then (Y ∩ Xℓ, E∗ℓω) must be trivial in
view of Example 3.5, i.e., Xℓ ⊂ Y and ω is constant on Xℓ, and hence (Y ∩Xm, E∗mω) is
by itself a relative 2e-design; cf. Remark 3.7. This shows that the above condition is not a
restrictive one. We also note that
Lemma 5.1. Let (Y, ω) be a relative t-design inQn supported on
⊔
ℓ∈LXℓ. Then (Y
′, Anω)
is a relative t-design supported on
⊔
ℓ∈LXn−ℓ, where Y
′ = {y′ : y ∈ Y }, and for every
y ∈ X , y′ denotes the unique vertex such that (y, y′) ∈ Rn.
Proof. Immediate from EiAn ∈ span{Ei} (0 ⩽ i ⩽ n).
In view of the above comments, we now make the following assumption:
Assumption 5.2. In this section, let (Y, ω) be a tight relative 2e-design (e ∈ N) in Qn
supported on two shells Xℓ ⊔Xm, where
e ⩽ ℓ < m ⩽ n− ℓ (⩽ n− e).
Our aim is to show that Y then induces the structure of a coherent configuration with
two fibers, and to obtain a necessary condition on the existence of such (Y, ω) akin to
Delsarte’s theorem on tight 2e-designs. To this end, we first recall the proof of (3.4) given
in [6, Theorem 2.7, Example 2.9] under the above assumption.
For convenience, set
E∗L = E
∗
ℓ + E
∗
m.
By (4.2) and (4.3), we have
E∗L
(
e∑
i=0
EiV
)
=
e∑
r=0
∑
W∈Λr
E∗L
(
e∑
i=r
EiW
)
. (5.1)
Let W ∈ Λr, where 0 ⩽ r ⩽ e. Recall Theorem 4.3 and also the standard basis (4.5) of
W . If r = e then EeW is spanned by vW , and hence we have
E∗LEeW = span{E∗LvW }.
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 181
Note that E∗LvW is non-zero by Assumption 5.2, and hence
dimE∗LEeW = 1
in this case. Suppose next that 0 ⩽ r < e. On the one hand, since
E∗L
(
e∑
i=r
EiW
)
⊂ E∗LW = E∗ℓW + E∗mW,
we have
dimE∗L
(
e∑
i=r
EiW
)
⩽ 2.
On the other hand, it follows from (2.8) that
vW , A
∗
1vW ∈ ErW + Er+1W ⊂
e∑
i=r
EiW, (5.2)
and hence
E∗LvW , E
∗
LA
∗
1vW ∈ E∗L
(
e∑
i=r
EiW
)
.
Moreover, we have (cf. (2.7))
E∗LvW = E
∗
ℓ vW + E
∗
mvW , E
∗
LA
∗
1vW = θ
∗
ℓE
∗
ℓ vW + θ
∗
mE
∗
mvW ,
so that these two vectors are non-zero and are linearly independent by Assumption 5.2 and
since θ∗ℓ ̸= θ∗m. It follows that
dimE∗L
(
e∑
i=r
EiW
)
= 2.
Note that in this case we in fact have
E∗L
(
e∑
i=r
EiW
)
= span{E∗ℓ vW , E∗mvW },
as
E∗ℓ vW = E
∗
L
θ∗mI −A∗1
θ∗m − θ∗ℓ
vW , E
∗
mvW = E
∗
L
θ∗ℓ I −A∗1
θ∗ℓ − θ∗m
vW . (5.3)
Combining these comments, we now obtain (3.4) as follows:
dimE∗L
(
e∑
i=0
EiV
)
=
e∑
r=0
∑
W∈Λr
dimE∗L
(
e∑
i=r
EiW
)
= |Λe|+
e−1∑
r=0
2|Λr|
= dimEeV + dimEe−1V
182 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
=
(
n
e
)
+
(
n
e− 1
)
,
where we have used (3.5) and (4.4).
By the above discussions, the set of vectors below forms an orthogonal basis of the
subspace (5.1): (
e−1⊔
r=0
⊔
W∈Λr
{E∗ℓ vW , E∗mvW }
)⊔( ⊔
W∈Λe
{E∗LvW }
)
.
As in the proof of Theorem 3.14, let
D = diagω.
We next apply
√
D to the above basis vectors and compute their inner products. First, let
W,W ′ ∈
⊔e−1
r=0 Λr. It is clear that〈√
DE∗ℓ vW ,
√
DE∗mvW ′
〉
=
〈√
DE∗mvW ,
√
DE∗ℓ vW ′
〉
= 0. (5.4)
By (5.3), we have
(E∗ℓ vW ) ◦ (E∗ℓ vW ′) = E∗Lu,
where means complex conjugate, and
u =
(
θ∗mI −A∗1
θ∗m − θ∗ℓ
vW
)
◦
(
θ∗mI −A∗1
θ∗m − θ∗ℓ
vW ′
)
.
Observe that u belongs to
∑2e
i=0EiV by (2.5) (applied to h = k = e) and (5.2). Hence, by
Proposition 3.6 we have〈√
DE∗ℓ vW ,
√
DE∗ℓ vW ′
〉
= ⟨ω,E∗Lu⟩
= ⟨ω, u⟩
=
⟨ω, X̂ℓ⟩
|Xℓ|
⟨X̂ℓ, u⟩+
⟨ω, X̂m⟩
|Xm|
⟨X̂m, u⟩
=
⟨ω, X̂ℓ⟩
|Xℓ|
⟨X̂ℓ, E∗Lu⟩
=
⟨ω, X̂ℓ⟩
|Xℓ|
⟨E∗ℓ vW , E∗ℓ vW ′⟩
= δW,W ′
⟨ω, X̂ℓ⟩
|Xℓ|
∥E∗ℓ vW ∥2. (5.5)
Likewise, we have〈√
DE∗mvW ,
√
DE∗mvW ′
〉
= δW,W ′
⟨ω, X̂m⟩
|Xm|
∥E∗mvW ∥2. (5.6)
Next, let W ∈
⊔e−1
r=0 Λr and W
′ ∈ Λe. Then, by the same argument we have〈√
DE∗ℓ vW ,
√
DE∗LvW ′
〉
=
〈√
DE∗mvW ,
√
DE∗LvW ′
〉
= 0. (5.7)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 183
Finally, let W,W ′ ∈ Λe. In this case, we have〈√
DE∗LvW ,
√
DE∗LvW ′
〉
= δW,W ′
(
⟨ω, X̂ℓ⟩
|X̂ℓ|
∥E∗ℓ vW ∥2 +
⟨ω, X̂m⟩
|X̂m|
∥E∗mvW ∥2
)
. (5.8)
Since (Y, ω) is a tight relative 2e-design, it follows from (5.4) – (5.8) that the set of vectors
below is an orthogonal basis of the subspace
√
DV = span{ŷ : y ∈ Y } of dimension
|Y | =
(
n
e
)
+
(
n
e−1
)
:(
e−1⊔
r=0
⊔
W∈Λr
{√
DE∗ℓ vW ,
√
DE∗mvW
})⊔( ⊔
W∈Λe
{√
DE∗LvW
})
.
For convenience, set
Yℓ = Y ∩Xℓ, Ym = Y ∩Xm.
We will naturally make the following identification by discarding irrelevant entries:
√
DE∗ℓ V = span{ŷ : y ∈ Yℓ} ←→ CYℓ ,√
DE∗mV = span{ŷ : y ∈ Ym} ←→ CYm .
We write
Λr =
{
W 1r ,W
2
r , . . . ,W
|Λr|
r
}
(0 ⩽ r ⩽ e).
For 0 ⩽ r ⩽ e, define a |Yℓ| × |Λr| matrix Hℓr and a |Ym| × |Λr| matrix Hmr by
Hℓr =
[√
DE∗ℓ vW 1r · · ·
√
DE∗ℓ vW |Λr|r
]
,
Hmr =
[√
DE∗mvW 1r · · ·
√
DE∗mvW |Λr|r
]
.
We then define a characteristic matrix H of (Y, ω) by
H =
[
Hℓ0 · · · Hℓe−1 O · · · O Hℓe
O · · · O Hm0 · · · Hme−1 Hme
]
.
We note that H is a square matrix of size |Y | =
(
n
e
)
+
(
n
e−1
)
. By (4.6), (4.7), and
(5.4) – (5.8), and since
|Xi| =
(
n
i
)
(0 ⩽ i ⩽ n),
we have
H†H =
(
e−1
⊕
r=0
κℓrI|Λr|
)
⊕
(
e−1
⊕
r=0
κmr I|Λr|
)
⊕ κeI|Λe|, (5.9)
where
κℓr =
ωℓ
(
n−2r
ℓ−r
)
2n−2r
(
n
ℓ
) , κmr = ωm
(
n−2r
m−r
)
2n−2r
(
n
m
) (0 ⩽ r < e),
184 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
κe =
ωℓ
(
n−2e
ℓ−e
)
2n−2e
(
n
ℓ
) + ωm(n−2em−e)
2n−2e
(
n
m
) ,
and we abbreviate
ωℓ = ⟨ω, X̂ℓ⟩, ωm = ⟨ω, X̂m⟩.
Let K denote the diagonal matrix on the RHS in (5.9). Then it follows that
I|Y | = HK
−1H†
=
[ ∑e
r=0
1
κℓr
Hℓr(H
ℓ
r)
† 1
κe
Hℓe(H
m
e )
†
1
κe
Hme (H
ℓ
e)
† ∑e
r=0
1
κmr
Hmr (H
m
r )
†
]
, (5.10)
where we write
κℓe = κ
m
e := κe (5.11)
for brevity. In particular, we have
1
κe
Hℓe(H
m
e )
† = O. (5.12)
Moreover, from (5.9) and (5.10) it follows that(
1
κℓr
Hℓr(H
ℓ
r)
†
)(
1
κℓr′
Hℓr′(H
ℓ
r′)
†
)
= δr,r′
1
κℓr
Hℓr(H
ℓ
r)
† (0 ⩽ r, r′ < e), (5.13)
1
κe
Hℓe(H
ℓ
e)
† = I|Yℓ| −
e−1∑
r=0
1
κℓr
Hℓr(H
ℓ
r)
†, (5.14)(
1
κmr
Hmr (H
m
r )
†
)(
1
κmr′
Hmr′ (H
m
r′ )
†
)
= δr,r′
1
κmr
Hmr (H
m
r )
† (0 ⩽ r, r′ < e), (5.15)
1
κe
Hme (H
m
e )
† = I|Ym| −
e−1∑
r=0
1
κmr
Hmr (H
m
r )
†. (5.16)
Note that the matrices (κℓr)
−1Hℓr(H
ℓ
r)
†, (κmr )
−1Hmr (H
m
r )
† (0 ⩽ r < e) are non-zero since
Hℓr , H
m
r are non-zero. Likewise, by setting
κr =
√
κℓrκ
m
r (0 ⩽ r < e)
for brevity, we have(
1
κℓr
Hℓr(H
ℓ
r)
†
)(
1
κr′
Hℓr′(H
m
r′ )
†
)
= δr,r′
1
κr
Hℓr(H
m
r )
† (0 ⩽ r, r′ < e), (5.17)(
1
κr
Hℓr(H
m
r )
†
)(
1
κmr′
Hmr′ (H
m
r′ )
†
)
= δr,r′
1
κr
Hℓr(H
m
r )
† (0 ⩽ r, r′ < e), (5.18)(
1
κr
Hℓr(H
m
r )
†
)(
1
κr′
Hmr′ (H
ℓ
r′)
†
)
= δr,r′
1
κℓr
Hℓr(H
ℓ
r)
† (0 ⩽ r, r′ < e), (5.19)(
1
κr
Hmr (H
ℓ
r)
†
)(
1
κr′
Hℓr′(H
m
r′ )
†
)
= δr,r′
1
κmr
Hmr (H
m
r )
† (0 ⩽ r, r′ < e). (5.20)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 185
Since the matrices (κℓr)
−1Hℓr(H
ℓ
r)
†, (κmr )
−1Hmr (H
m
r )
† (0 ⩽ r < e) are non-zero, it fol-
lows from (5.17) – (5.20) that the matrices (κr)−1Hℓr(H
m
r )
† (0 ⩽ r < e) are non-zero and
are linearly independent.
It follows from Theorem 3.14 and Proposition 4.1 that ω is constant on each of Yℓ and
Ym, from which it follows that
Dy,y = ω(y) =

ωℓ
|Yℓ|
if y ∈ Yℓ,
ωm
|Ym|
if y ∈ Ym.
(5.21)
Hence, by comparing with the formula (4.8) for the matrices Ĕi,jr , we have
1
κℓr
Hℓr(H
ℓ
r)
† =
(
n
ℓ
)
|Yℓ|
Ĕℓ,ℓr |Yℓ×Yℓ (0 ⩽ r < e), (5.22)
1
κe
Hℓe(H
ℓ
e)
† =
ωℓ
(
n−2e
ℓ−e
)
2n−2eκe|Yℓ|
Ĕℓ,ℓe |Yℓ×Yℓ , (5.23)
1
κmr
Hmr (H
m
r )
† =
(
n
m
)
|Ym|
Ĕm,mr |Ym×Ym (0 ⩽ r < e), (5.24)
1
κe
Hme (H
m
e )
† =
ωm
(
n−2e
m−e
)
2n−2eκe|Ym|
Ĕm,me |Ym×Ym , (5.25)
1
κr
Hℓr(H
m
r )
† =
√(
n
ℓ
)(
n
m
)√
|Yℓ||Ym|
Ĕℓ,mr |Yℓ×Ym (0 ⩽ r < e), (5.26)
1
κe
Hℓe(H
m
e )
† =
√
ωℓωm
(
n−2e
ℓ−e
)(
n−2e
m−e
)
2n−2eκe
√
|Yℓ||Ym|
Ĕℓ,me |Yℓ×Ym , (5.27)
where |Yℓ×Yℓ etc. mean taking corresponding submatrices. From (5.23) and (5.25) it fol-
lows that the matrices (κe)−1Hℓe(H
ℓ
e)
†, (κe)
−1Hme (H
m
e )
† are also non-zero, since each of
Ĕℓ,ℓe |Yℓ×Yℓ , Ĕm,me |Ym×Ym has non-zero constant diagonal entries by (4.14).
Let H ′ be the set consisting of the |Y | × |Y | matrices of the form[ ∑e
r=0 a
ℓ,ℓ
r
1
κℓr
Hℓr(H
ℓ
r)
† ∑e−1
r=0 a
ℓ,m
r
1
κr
Hℓr(H
m
r )
†∑e−1
r=0 a
m,ℓ
r
1
κr
Hmr (H
ℓ
r)
† ∑e
r=0 a
m,m
r
1
κmr
Hmr (H
m
r )
†
]
,
where aℓ,ℓr etc. are in C, and we are again using the notation (5.11). By (5.13) – (5.20) and
the above comments, H ′ is a C-algebra with
dimH ′ = 4e+ 2. (5.28)
Define
Sℓ,ℓ(Y ) =
{
j : Rj ∩ (Yℓ × Yℓ) ̸= ∅
}
,
and define Sℓ,m(Y )(= Sm,ℓ(Y )) and Sm,m(Y ) in the same manner. Let H be the set
consisting of the |Y | × |Y | matrices of the form[ ∑
j∈Sℓ,ℓ(Y ) b
ℓ,ℓ
j Aj |Yℓ×Yℓ
∑
j∈Sℓ,m(Y ) b
ℓ,m
j Aj |Yℓ×Ym∑
j∈Sm,ℓ(Y ) b
m,ℓ
j Aj |Ym×Yℓ
∑
j∈Sm,m(Y ) b
m,m
j Aj |Ym×Ym
]
, (5.29)
186 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
where bℓ,ℓj etc. are in C. Then H is a C-vector space with
dimH = |Sℓ,ℓ(Y )|+ |Sℓ,m(Y )|+ |Sm,ℓ(Y )|+ |Sm,m(Y )|. (5.30)
Note that H is closed under ◦. By (5.22) – (5.26) and Proposition 4.1 (or (4.14)), H ′ is a
subspace of H .
By (4.14), (5.14), (5.22), and (5.23), we have
I|Yℓ| =
e−1∑
r=0
(
n
ℓ
)
|Yℓ|
Ĕℓ,ℓr |Yℓ×Yℓ +
ωℓ
(
n−2e
ℓ−e
)
2n−2eκe|Yℓ|
Ĕℓ,ℓe |Yℓ×Yℓ
=
1
|Yℓ|
min{ℓ,n−ℓ}∑
ξ=0
 e−1∑
r=0
((
n
r
)
−
(
n
r − 1
))
3F2
(
−ξ,−r, r − n− 1
ℓ− n,−ℓ
∣∣∣∣ 1)
+
ωℓ
(
n−2e
ℓ−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
(
n
ℓ
) 3F2(−ξ,−e, e− n− 1ℓ− n,−ℓ
∣∣∣∣ 1)
A2ξ|Yℓ×Yℓ . (5.31)
Hence it follows that {ξ ̸= 0 : 2ξ ∈ Sℓ,ℓ(Y )} is a set of zeros of the polynomial
ψℓ,ℓe (ξ) =
e−1∑
r=0
((
n
r
)
−
(
n
r − 1
))
3F2
(
−ξ,−r, r − n− 1
ℓ− n,−ℓ
∣∣∣∣ 1)
+
ωℓ
(
n−2e
ℓ−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
(
n
ℓ
) 3F2(−ξ,−e, e− n− 1ℓ− n,−ℓ
∣∣∣∣ 1) ∈ R[ξ]. (5.32)
Note that ψℓ,ℓe (ξ) has degree exactly e, from which it follows that
|Sℓ,ℓ(Y )| ⩽ e+ 1. (5.33)
Likewise, we find that {ξ ̸= 0 : 2ξ ∈ Sm,m(Y )} is a set of zeros of the polynomial
ψm,me (ξ) =
e−1∑
r=0
((
n
r
)
−
(
n
r − 1
))
3F2
(
−ξ,−r, r − n− 1
m− n,−m
∣∣∣∣ 1)
+
ωm
(
n−2e
m−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
(
n
m
) 3F2(−ξ,−e, e− n− 1m− n,−m
∣∣∣∣ 1) ∈ R[ξ], (5.34)
and hence that
|Sm,m(Y )| ⩽ e+ 1. (5.35)
Finally, by (4.14), (5.12), and (5.27), we have
O =
√
ωℓωm
(
n−2e
ℓ−e
)(
n−2e
m−e
)
2n−2eκe
√
|Yℓ||Ym|
Ĕℓ,me |Yℓ×Ym
=
√
ωℓωm
(
n−m
e
)(
n−2e
ℓ−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
√
|Yℓ||Ym|
(
n
ℓ
)(
n−ℓ
e
)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 187
×
min{ℓ,n−m}∑
ξ=0
3F2
(
−ξ,−e, e− n− 1
m− n,−ℓ
∣∣∣∣ 1)A2ξ+m−ℓ|Yℓ×Ym .
Hence it follows that {ξ : 2ξ +m− ℓ ∈ Sℓ,m(Y )} is a set of zeros of the polynomial
ψℓ,me (ξ) = 3F2
(
−ξ,−e, e− n− 1
m− n,−ℓ
∣∣∣∣ 1) ∈ R[ξ], (5.36)
and that
|Sℓ,m(Y )| = |Sm,ℓ(Y )| ⩽ e. (5.37)
By (5.30), (5.33), (5.35), and (5.37), we have
dimH ⩽ 4e+ 2.
Since H ′ is a subspace of H , it follows from (5.28) that H = H ′. In particular, H is a
C-algebra. It is also clear that H is closed under † and contains J|Y |. We now conclude
that H is a coherent algebra. Note also that equality holds in each of (5.33), (5.35), and
(5.37).
To summarize:
Theorem 5.3. Recall Assumption 5.2. With the above notation, the following hold:
(i) The set H from (5.29) is a coherent algebra of type
[
e+1 e
e e+1
]
.
(ii) The sets of zeros of the polynomials ψℓ,ℓe (ξ), ψ
m,m
e (ξ), and ψ
ℓ,m
e (ξ) from (5.32),
(5.34), and (5.36) are given respectively by
{ξ ̸= 0 : 2ξ ∈ Sℓ,ℓ(Y )}, {ξ ̸= 0 : 2ξ ∈ Sm,m(Y )}, and
{ξ : 2ξ +m− ℓ ∈ Sℓ,m(Y )}.
In particular, the zeros of these polynomials are integral.
Concerning the scalars ωℓ and ωm appearing in the polynomials ψℓ,ℓe (ξ) and ψ
m,m
e (ξ),
it follows that
Proposition 5.4. Recall Assumption 5.2. The scalars ωℓ and ωm satisfies
ωm
ωℓ
=
(
n
m
)(
n−2e
ℓ−e
)(
n
ℓ
)(
n−2e
m−e
) · |Ym| − ( ne−1)
|Yℓ| −
(
n
e−1
) .
In particular, the weight function ω is unique up to a scalar multiple.
Proof. By comparing the diagonal entries of both sides in (5.31), we have
1 =
ψℓ,ℓe (0)
|Yℓ|
=
1
|Yℓ|
( n
e− 1
)
+
ωℓ
(
n−2e
ℓ−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
(
n
ℓ
)
 .
Likewise,
1 =
ψm,me (0)
|Ym|
=
1
|Ym|
( n
e− 1
)
+
ωm
(
n−2e
m−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
(
n
m
)
 .
By eliminating κe, we obtain the formula for ωm(ωℓ)−1. The uniqueness of ω follows from
this and (5.21).
188 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
Example 5.5. Suppose that e = 1. In this case, Theorem 5.3(i) was previously obtained by
Bannai, Bannai, and Bannai [5, Theorem 2.2 (i)]. Moreover, Theorem 5.3(ii) and Proposi-
tion 5.4 are together equivalent to [5, Proposition 4.3].
Example 5.6. Suppose that e = 2. Then we have
ψℓ,me (ξ) = 1 +
(−ξ)(−2)(1− n)
(m− n)(−ℓ)
+
(−ξ)(1− ξ)(−2)(−1)(1− n)(2− n)
(m− n)(m− n+ 1)(−ℓ)(1− ℓ)2
= 1− 2(n− 1)ξ
(n−m)ℓ
+
(n− 1)(n− 2)ξ(ξ − 1)
(n−m)(n−m− 1)ℓ(ℓ− 1)
.
From Example 3.13 we find two parameter sets satisfying Assumption 5.2:
n ℓ m ξ
22 6 7 3, 5
22 6 15 1, 3
The zeros ξ given in the last column are indeed integers. Note that the other two parameter
sets in Example 3.13 correspond to the complements of these two; cf. Lemma 5.1. On the
other hand, the existence of tight relative 4-designs with the following feasible parameter
sets was left open in [9, Section 6]:
n ℓ m ξ
37 9 16 114 (71±
√
337)
37 9 21 114 (55±
√
337)
41 15 16 126 (237±
√
1569)
41 15 25 126 (153±
√
1569)
Here, we are again taking Lemma 5.1 into account. Observe that the zeros ξ are irrational,
thus proving the non-existence.
We end this section with a comment on the expressions of the polynomials ψℓ,ℓe (ξ)
and ψm,me (ξ). We first invoke the following identity which agrees with the formula of the
backward shift operator on the dual Hahn polynomials (cf. [31, Section 1.6]):
α(N + 1)(α+ β + 2r)Qr(ξ;α− 1, β,N + 1)
= (α+ r)(α+ β + r)(N + 1− r)Qr(ξ − 1;α, β,N)
− r(α+ β +N + 1 + r)(β + r)Qr−1(ξ − 1;α, β,N). (5.38)
This can be routinely verified by writing the LHS as a linear combination of the polynomi-
als (1− ξ)i (0 ⩽ i ⩽ r) using
(−ξ)i = (1− ξ)i − i(1− ξ)i−1,
and then comparing the coefficients of both sides. Setting α = ℓ − n, β = −ℓ − 1, and
N = ℓ − 1 in (5.38), it follows that the first term of the RHS in (5.32) is rewritten as
follows:
e−1∑
r=0
((
n
r
)
−
(
n
r − 1
))
3F2
(
−ξ,−r, r − n− 1
ℓ− n,−ℓ
∣∣∣∣ 1)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 189
=
e−1∑
r=0
n!(n− 2r + 1)
r!(n− r + 1)!
Qr(ξ;α− 1, β,N + 1)
=
n!
ℓ(n− ℓ)
e−1∑
r=0
(
(ℓ− n+ r)(r − n− 1)(ℓ− r)
r!(n− r + 1)!
Qr(ξ − 1;α, β,N)
−r(r + ℓ− n− 1)(r − ℓ− 1)
r!(n− r + 1)!
Qr−1(ξ − 1;α, β,N)
)
=
n!
ℓ(n− ℓ)
· (−1)(ℓ− n+ e− 1)(ℓ− e+ 1)
(e− 1)!(n− e+ 1)!
Qe−1(ξ − 1;α, β,N)
=
(
n
e− 1
)
(n− ℓ− e+ 1)(ℓ− e+ 1)
ℓ(n− ℓ) 3
F2
(
1− ξ, 1− e, e− n− 1
ℓ− n+ 1, 1− ℓ
∣∣∣∣ 1) .
Likewise, the first term of the RHS in (5.34) is given by
e−1∑
r=0
((
n
r
)
−
(
n
r − 1
))
3F2
(
−ξ,−r, r − n− 1
m− n,−m
∣∣∣∣ 1)
=
(
n
e− 1
)
(n−m− e+ 1)(m− e+ 1)
m(n−m) 3
F2
(
1− ξ, 1− e, e− n− 1
m− n+ 1, 1−m
∣∣∣∣ 1) .
6 Zeros of the Hahn and Hermite polynomials
Recall the Hahn polynomials Qr(ξ;α, β,N) from (4.11). Recall also that the zeros of
orthogonal polynomials are always real and simple; see, e.g., [42, Theorem 3.3.1]. It is
well known that we can obtain the Hermite polynomials as limits of the Hahn polynomials;
cf. [30, 31]. In this section, we revisit this limit process and describe the limit behavior of
the zeros of the Qr(ξ;α, β,N), in a special case which is suited to our purpose.
Assumption 6.1. Throughout this section, we assume that α < −N and β < −N , so
that the Qr(ξ;α, β,N) satisfy the orthogonality relation (4.12). We consider the following
limit:
ϵ := − α+ β√
αβN
→ +0.
We write
α =
αϵ
ϵ2
, β =
βϵ
ϵ2
, N =
Nϵ
ϵ2
,
and assume further that
lim
ϵ→+0
Nϵ
αϵ + βϵ
= 0, lim
ϵ→+0
βϵ
αϵ + βϵ
= ρ ∈ [0, 1].
Remark 6.2. We do not require in Assumption 6.1 that αϵ, βϵ, and Nϵ are uniquely deter-
mined by ϵ. In other words, these are multi-valued functions of ϵ in general (for admissible
values of ϵ), but their limit behaviors are uniformly governed by ϵ.
With reference to Assumption 6.1, observe that
lim
ϵ→+0
αϵ = lim
ϵ→+0
αϵ2 = lim
ϵ→+0
αϵ + βϵ
βϵ
· αϵ + βϵ
Nϵ
= −∞.
190 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
Likewise, we have
lim
ϵ→+0
βϵ = −∞, lim
ϵ→+0
Nϵ =
1
ρ(1− ρ)
∈ [4,∞].
We will work with the normalized (or monic) Hahn polynomials:
qr(ξ) = qr(ξ; ϵ) =
(α+ 1)r(−N)r
(r + α+ β + 1)r
Qr(ξ;α, β,N). (6.1)
Their recurrence relation is given by (cf. [31, Section 1.5])
ξqr(ξ) = qr+1(ξ) + (ar + br)qr(ξ) + ar−1brqr−1(ξ), (6.2)
where q−1(ξ) := 0, and
ar =
(r + α+ β + 1)(r + α+ 1)(N − r)
(2r + α+ β + 1)(2r + α+ β + 2)
,
br =
r(r + α+ β +N + 1)(r + β)
(2r + α+ β)(2r + α+ β + 1)
.
For convenience, let
λϵ =
√
2(αϵ + βϵ +Nϵ)
αϵ + βϵ
.
Note that
lim
ϵ→+0
λϵ =
√
2. (6.3)
Consider the polynomial q̃r(η; ϵ) in the new indeterminate η defined by
q̃r(η) = q̃r(η; ϵ) = qr
(
λϵη
ϵ
+
αϵNϵ
(αϵ + βϵ)ϵ2
)
· ϵ
r
(λϵ)r
∈ R[η].
Note that q̃r(η) is also monic with degree r in η. Then (6.2) becomes
ηq̃r(η) = q̃r+1(η) +
1
λϵ
(
(ar + br)ϵ−
αϵNϵ
(αϵ + βϵ)ϵ
)
q̃r(η) +
ar−1brϵ
2
(λϵ)2
q̃r−1(η). (6.4)
It is a straightforward matter to show that
1
λϵ
(
(ar + br)ϵ−
αϵNϵ
(αϵ + βϵ)ϵ
)
= −(µϵ + rσϵ)ϵ+O(ϵ3), (6.5)
ar−1brϵ
2
(λϵ)2
=
r
2
+O(ϵ2), (6.6)
where
µϵ :=
(αϵ − βϵ)Nϵ
λϵ(αϵ + βϵ)2
, σϵ :=
(αϵ − βϵ)(αϵ + βϵ + 2Nϵ)
λϵ(αϵ + βϵ)2
are convergent:
lim
ϵ→+0
µϵ = 0, lim
ϵ→+0
σϵ =
1− 2ρ√
2
. (6.7)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 191
Recall the Hermite polynomials [31, Section 1.13]
Hr(η) = (2η)
r
2F0
(
−r/2,−(r − 1)/2
−
∣∣∣∣− 1η2
)
∈ R[η] (r = 0, 1, 2, . . .).
Their normalized recurrence relation is given by
ηhr(η) = hr+1(η) +
r
2
hr−1(η), (6.8)
where
hr(η) =
Hr(η)
2r
, (6.9)
and h−1(η) := 0. We also note that
dhr
dη
(η) = rhr−1(η), (6.10)
and that
hr(−η) = (−1)rhr(η). (6.11)
Since q̃0(η) = h0(η) = 1, it follows from (6.4) – (6.8) that
lim
ϵ→+0
q̃r(η; ϵ) = hr(η) (6.12)
in the sense of coefficient-wise convergence.
We now set
q̃r(η; 0) = hr(η),
and discuss partial derivatives of q̃r(η; ϵ) as a bivariate function of η and ϵ. First, it follows
from (6.10) and (6.12) that
lim
ϵ→+0
∂q̃r
∂η
(η; ϵ) =
dhr
dη
(η) = rhr−1(η). (6.13)
Concerning the partial differentiability of q̃r(η; ϵ) with respect to ϵ, it follows that
Lemma 6.3. The function q̃r(η; ϵ) is partially right differentiable with respect to ϵ at (η, 0),
and we have
∂q̃r
∂ϵ
(η; 0) =
r(1− 2ρ)
3
√
2
(
(r − 1 + η2)hr−1(η)− ηhr(η)
)
.
Proof. Throughout the proof, we fix η ∈ R and set
∆r(ϵ) = ∆r(η; ϵ) =
q̃r(η; ϵ)− hr(η)
ϵ
.
It follows from (6.4) – (6.8) and (6.12) that
η∆r(ϵ) = ∆r+1(ϵ)− (µϵ + rσϵ)q̃r(η; ϵ) +
r
2
∆r−1(ϵ) +O(ϵ)
= ∆r+1(ϵ)− rσ0hr(η) +
r
2
∆r−1(ϵ) + o(1), (6.14)
192 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
where we set
σ0 := lim
ϵ→+0
σϵ =
1− 2ρ√
2
for brevity. Since q̃0(η; ϵ) = 1, we have ∆0(ϵ) = 0. Solving the recurrence (6.14) using
this initial condition and (6.8), we routinely obtain
∆r(ϵ) =
r(r − 1)
2
σ0hr−1(η) +
r(r − 1)(r − 2)
12
σ0hr−3(η) + o(1),
where h−1(η) = h−2(η) = h−3(η) := 0. It follows that q̃r(η; ϵ) is partially right differen-
tiable with respect to ϵ at (η, 0):
∂q̃r
∂ϵ
(η; 0) = lim
ϵ→+0
∆r(ϵ)
=
r(r − 1)
2
σ0hr−1(η) +
r(r − 1)(r − 2)
12
σ0hr−3(η).
Finally, from (6.8) it follows that
∂q̃r
∂ϵ
(η; 0) =
r(r − 1)
2
σ0hr−1(η) +
r(r − 1)
6
σ0
(
ηhr−2(η)− hr−1(η)
)
=
r(r − 1)
3
σ0hr−1(η) +
r
3
σ0η
(
ηhr−1(η)− hr(η)
)
=
rσ0
3
(
(r − 1 + η2)hr−1(η)− ηhr(η)
)
,
as desired.
Proposition 6.4. Recall Assumption 6.1. Fix a positive integer e, and let
ξ−⌊e/2⌋ < · · · < ξ−1 < (ξ0) < ξ1 < · · · < ξ⌊e/2⌋,
η−⌊e/2⌋ < · · · < η−1 < (η0) < η1 < · · · < η⌊e/2⌋
be the zeros of qe(ξ; ϵ) and he(η) from (6.1) and (6.9), respectively, where ξ0 and η0 appear
only when e is odd. Then ξi satisfies
lim
ϵ→+0
(
ξi −
λϵηi
ϵ
− αϵNϵ
(αϵ + βϵ)ϵ2
)
=
2ρ− 1
3
(
e− 1 + (ηi)2
)
as a function of ϵ, for i = −⌊e/2⌋, . . . ,−1, (0), 1, . . . , ⌊e/2⌋.
Proof. Define τi by
ξi =
λϵ(ηi + τi)
ϵ
+
αϵNϵ
(αϵ + βϵ)ϵ2
,
so that ηi + τi is a zero of q̃e(η; ϵ). Then, from (6.12) it follows that
lim
ϵ→+0
τi = 0. (6.15)
For the moment, fix i. Then we have
0 = q̃e(ηi + τi; ϵ) = q̃e(ηi; ϵ) +
∂q̃e
∂η
(ηi + θτi; ϵ)τi
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 193
for some θ ∈ (0, 1) depending on ϵ. Hence, from (6.13), (6.15), Lemma 6.3, and since
q̃e(ηi; 0) = he(ηi) = 0,
it follows that
lim
ϵ→+0
τi
ϵ
= − 1
ehe−1(ηi)
lim
ϵ→+0
q̃e(ηi; ϵ)
ϵ
= − 1
ehe−1(ηi)
∂q̃e
∂ϵ
(ηi; 0)
=
2ρ− 1
3
√
2
(
e− 1 + (ηi)2
)
,
where we note that he(η) and he−1(η) have no common zero by the general theory of
orthogonal polynomials; see, e.g., [42, Theorem 3.3.2]. By (6.3), we have
lim
ϵ→+0
(
ξi −
λϵηi
ϵ
− αϵNϵ
(αϵ + βϵ)ϵ2
)
= lim
ϵ→+0
λϵτi
ϵ
=
2ρ− 1
3
(
e− 1 + (ηi)2
)
.
This completes the proof.
The following is part of the estimates on the zeros of he(η) used in [1].3
Proposition 6.5 ([1, Proposition 13]). Fix a positive integer e, and let the ηi be as in
Proposition 6.4. Then η−i = −ηi for all i. Moreover, the following hold:
1. If e is odd and e ⩾ 5, then η0 = 0 and (η1)2 < 3/2.
2. If e is even and e ⩾ 8, then (η2)2 − (η1)2 < 3/2.
Proof. That η−i = −ηi is immediate from (6.11). We now write ηi = ηei to compare these
zeros for different values of e. Then, as an application of Sturm’s method, it follows that
√
2e+ 1 ηei <
√
2e′ + 1 ηe
′
i (i = 1, 2, . . . , ⌊e′/2⌋),
whenever e′ < e and e′ ≡ e (mod 2); see the comments preceding (6.31.19) in [42]. Since
h3(η) = η
3 − 3
2
η, h4(η) = η
4 − 3η2 + 3
4
,
we have
η31 =
√
3
2
, η42 =
√
3 +
√
6
2
.
Hence, for odd e ⩾ 5 we have
(ηe1)
2 <
7
2e+ 1
(η31)
2 =
21
4e+ 2
<
3
2
,
and for even e ⩾ 8 we have
(ηe2)
2 − (ηe1)2 < (ηe2)2 <
9
2e+ 1
(η42)
2 =
27 + 9
√
6
4e+ 2
<
3
2
,
as desired.
3Bannai [1] worked with the polynomial
√
2ehe(η/
√
2). We may remark that the upper bounds
√
3 men-
tioned in Proposition 13 (i) and (ii) in [1] should both be 3. See also [22, Proposition 2.4].
194 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
7 A finiteness result for tight relative 2e-designs on two shells in Qn
In this section, we prove that
Theorem 7.1. For any δ ∈ (0, 1/2), there exists e0 = e0(δ) > 0 with the property that,
for every given integer e ⩾ e0 and each constant c > 0, there are only finitely many tight
relative 2e-designs (Y, ω) (up to scalar multiples of ω) supported on two shells Xℓ ⊔Xm
in Qn satisfying Assumption 5.2 such that
ℓ < c · nδ. (7.1)
Our proof is an application of Bannai’s method from [1]. We will use the following
result, which is a variation of [40, Satz I]:
Proposition 7.2. For any ϑ > 0 and δ ∈ (0, 1/ϑ), there exists k0 = k0(ϑ, δ) > 0 such
that the following holds for every given integer k ⩾ k0 and each constant c > 0: for all but
finitely many pairs (a, b) of positive integers with
b < c · aδ,
the product of k consecutive odd integers
(2a+ 1)(2a+ 3) · · · (2a+ 2k− 1)
has a prime factor which is greater than 2k + 1 and whose exponent in this product is
greater than that in
(b+ 1)(b+ 2) · · · (b+ ⌊ϑk⌋).
The proof of Proposition 7.2 will be deferred to the appendix.
We will establish Theorem 7.1 by contradiction:
Assumption 7.3. We fix δ ∈ (0, 1/2). Let k0 = k0(2, δ) > 0 be as in Proposition 7.2
(applied to ϑ = 2), and set
e0 = e0(δ) = max{2k0, 8}.
We also fix a positive integer e ⩾ e0 and a constant c > 0. Throughout the proof, we
assume that there exist infinitely many tight relative 2e-designs (Y, ω) in question.
Let Θ denote the set of triples (ℓ,m, n) ∈ N3 taken by those (Y, ω) in Assumption 7.3.
Recall from Proposition 5.4 that ω is uniquely determined by Y up to a scalar multiple.
Moreover, for each (ℓ,m, n) ∈ Θ there are only finitely many choices for Y . Hence we
have
|Θ| =∞. (7.2)
For the moment, we fix (ℓ,m, n) ∈ Θ and consider the polynomial ψℓ,me (ξ) (which
also depends on n) from (5.36). We recall that
ψℓ,me (ξ) = Qe(ξ;α, β,N),
where
α = m− n− 1, β = −m− 1, N = ℓ. (7.3)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 195
We note that α, β < −N in view of Assumption 5.2. By Theorem 5.3(ii), if we let
ξ−⌊e/2⌋ < · · · < ξ−1 < (ξ0) < ξ1 < · · · < ξ⌊e/2⌋ (7.4)
denote the zeros of ψℓ,me (ξ) (cf. Proposition 6.4), then we have
ξi ∈ {0, 1, . . . , ℓ} for all i. (7.5)
We also rewrite ψℓ,me (ξ) as follows:
ψℓ,me (ξ) =
e∑
i=0
se−i(−1)i(−ξ)i,
where
se−i =
(
e
i
)
(e− n− 1)i
(m− n)i(−ℓ)i
(0 ⩽ i ⩽ e).
From (7.5) it follows that the polynomial ψℓ,me (ξ)/s0 is monic and integral:
ψℓ,me (ξ)
s0
=
e∑
i=0
se−i
s0
(−1)i(−ξ)i = (ξ − ξ−⌊e/2⌋) · · · (ξ − ξ⌊e/2⌋) ∈ Z[ξ], (7.6)
where the factor (ξ − ξ0) appears only when e is odd. Since (−1)i(−ξ)i is also monic and
integral, and has degree i for 0 ⩽ i ⩽ e, it follows that
si
s0
= (−1)i
(
e
i
)
(n−m− e+ 1)i(ℓ− e+ 1)i
(n− 2e+ 2)i
∈ Z\{0} (0 ⩽ i ⩽ e), (7.7)
where that these coefficients are non-zero follows from Assumption 5.2.
We now consider the map f : Θ→ [0, 1]2 defined by
f(ℓ,m, n) =
(
ℓ
n
,
m
n
)
∈ [0, 1]2 ((ℓ,m, n) ∈ Θ).
Recall (7.2). Moreover, from (7.1) it follows that
|f−1(a, b)| <∞ ((a, b) ∈ [0, 1]2). (7.8)
Hence it follows that
|f(Θ)| =∞,
so that f(Θ) has at least one accumulation point in [0, 1]2. Again by (7.1), such an accu-
mulation point must be of the form
(0, ρ) ∈ [0, 1]2.
We next show that the parameters α, β, and N from (7.3) satisfy Assumption 6.1 when
f(ℓ,m, n)→ (0, ρ).
Claim 7.4. ℓ,m, n−m→∞ as f(ℓ,m, n)→ (0, ρ).
196 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
Proof. Since m,n −m ⩾ ℓ by Assumption 5.2, it suffices to show that ℓ → ∞. Suppose
the contrary, i.e., that there is a sequence (ℓk,mk, nk) (k ∈ N) of distinct elements of Θ
such that
lim
k→∞
f(ℓk,mk, nk) = (0, ρ), sup
k
ℓk <∞.
Since the ℓk are bounded, it follows from (7.5) and (7.6) that there are only finitely many
choices for ψℓ,me (ξ)/s0 when (ℓ,m, n) ranges over this sequence. In particular, there are
only finitely many choices for each of the coefficients s1/s0 and s2/s0, and hence the same
is true (cf. (7.7)) for each of
n−m− e+ 1
n− 2e+ 2
,
n−m− e+ 2
n− 2e+ 3
.
However, it is immediate to see that these distinct scalars in turn determine n and m
uniquely, from which it follows that the nk are bounded, a contradiction.
Claim 7.5. ℓm(n−m)/n2 →∞ as f(ℓ,m, n)→ (0, ρ).
Proof. If 0 < ρ < 1 then the result follows from Claim 7.4 and since
m(n−m)
n2
→ ρ(1− ρ) > 0.
Suppose next that ρ = 1. Suppose moreover that there is a sequence (ℓk,mk, nk)
(k ∈ N) of distinct elements of Θ such that
lim
k→∞
f(ℓk,mk, nk) = (0, 1), sup
k
ℓkmk(nk −mk)
(nk)2
<∞.
Since mk/nk → 1, we then have
sup
k
ℓk(nk −mk)
nk
<∞.
Let
rk =
(nk −mk − e+ 1)(ℓk − e+ 1)
nk − 2e+ 2
, tk =
(nk −mk − e+ 2)(ℓk − e+ 2)
nk − 2e+ 3
.
Then the rk and the tk are bounded since
rk ≈ tk ≈
ℓk(nk −mk)
nk
by Claim 7.4. From (7.7) it follows that s1/s0 and s2/s0 are bounded as well, and hence
take only finitely many non-zero integral values when (ℓ,m, n) ranges over this sequence.
It follows that the rk and the tk can assume only finitely many values, and then since
rk ≈ tk we must have rk = tk for sufficiently large k. However, it is again immediate to
see that rk ̸= tk for every k ∈ N, and hence this is absurd. It follows that the result holds
when ρ = 1.
Finally, suppose that ρ = 0. For every (ℓ,m, n) ∈ Θ we have
e
(m− e+ 1)(ℓ− e+ 1)
n− 2e+ 2
=
s1
s0
+ e(ℓ− e+ 1),
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 197
(
e
2
)
(m− e+ 1)2(ℓ− e+ 1)2
(n− 2e+ 2)2
=
s2
s0
+ (e− 1)(ℓ− e+ 2)s1
s0
+
(
e
2
)
(ℓ− e+ 1)2.
From (7.7) and Assumption 5.2 it follows that these scalars are non-zero integers. By the
same argument as above, but working with these two scalars instead of s1/s0 and s2/s0,
we conclude that the result holds in this case as well.
By Claims 7.4 and 7.5, it follows that the parameters α, β, and N from (7.3) satisfy
Assumption 6.1 when f(ℓ,m, n)→ (0, ρ), since
− α+ β√
αβN
≈ n√
ℓm(n−m)
,
N
α+ β
≈ − ℓ
n
,
β
α+ β
≈ m
n
.
Note that the scalar ρ in Assumption 6.1 agrees with the one used here in this case. Hence
we are now in the position to apply the results of the previous section to ψℓ,me (ξ), which is
the Hahn polynomial having these parameters.
Claim 7.6. We have ρ = 1/2. In particular, (0, 1/2) is a unique accumulation point of
f(Θ). Moreover, we have n = 2m for all but finitely many (ℓ,m, n) ∈ Θ.
Proof. Let the ξi be as in (7.4). Then from Propositions 6.4 and 6.5 it follows that
ξi + ξ−i − ξj − ξ−j →
4ρ− 2
3
(
(ηi)
2 − (ηj)2
)
for all i, j, (7.9)
as f(ℓ,m, n)→ (0, ρ), where the ηi are the zeros of the monic Hermite polynomial he(η)
from (6.9) as in Proposition 6.4. Recall that e ⩾ 8 by Assumption 7.3. Set (i, j) = (1, 0)
in (7.9) if e is odd, and (i, j) = (2, 1) if e is even. Then, since∣∣∣∣4ρ− 23
∣∣∣∣ ⩽ 23 ,
it follows from Proposition 6.5 that the RHS in (7.9) lies in the open interval (−1, 1).
However, the LHS in (7.9) is always an integer by (7.5), so that this is possible only when
the RHS equals zero, i.e., ρ = 1/2. In particular, we have shown that (0, 1/2) is a unique
accumulation point of f(Θ).
Again by (7.5) and (7.9), we then have
ξi + ξ−i = ξj + ξ−j for all i, j,
provided that f(ℓ,m, n) is sufficiently close to (0, 1/2). By the uniqueness of the accu-
mulation point and (7.8), this last condition on f(ℓ,m, n) can be rephrased as “for all but
finitely many (ℓ,m, n) ∈ Θ.” Now, let ξ̃ be the average of the zeros ξi of ψℓ,me (ξ). Then
the above identity means that the ξi are symmetric with respect to ξ̃. Hence, if we write
ψℓ,me (ξ)
s0
=
e∑
i=0
we−i(ξ − ξ̃)i,
then we have
w2i−1 = 0 (1 ⩽ i ⩽ ⌈e/2⌉)
198 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
for all but finitely many (ℓ,m, n) ∈ Θ. On the other hand, using (7.6) and (7.7), we
routinely obtain
w3 =
(
e
3
)
(n− 2ℓ)(n− 2m)
× (ℓ− e+ 1)(m− e+ 1)(n− ℓ− e+ 1)(n−m− e+ 1)
(n− 2e+ 2)3(n− 2e+ 3)(n− 2e+ 4)
.
Hence, by Assumption 5.2, that w3 = 0 forces n = 2m. The claim is proved.
By virtue of Claim 7.6, we may now assume without loss of generality that
n = 2m ((ℓ,m, n) ∈ Θ),
by discarding a finite number of exceptions. Set
k =
⌊e
2
⌋
,
and let c′ be a constant such that c′ > 2δc. Note that
k ⩾ k0 = k0(2, δ)
by Assumption 7.3. Let (ℓ,m, 2m) ∈ Θ. We have
c · (2m)δ < c′ · (m− e+ 1)δ
provided that m is large. Hence it follows from Proposition 7.2 (applied to ϑ = 2) and
(7.1) that if m is sufficiently large then there is a prime p > 2k+ 1 such that
νp((2m− 2e+ 3)(2m− 2e+ 5) · · · (2m− 2e+ 2k+ 1)) > νp((ℓ− e+ 1)2k),
where νp(n) denotes the exponent of p in n. Assuming that this is the case, let i (1 ⩽ i ⩽ k)
be such that
νp(2m− 2e+ 2i+ 1) > 0.
Observe that i is unique since p > 2k+ 1, so that we have
νp(2m− 2e+ 2i+ 1) > νp((ℓ− e+ 1)2k).
Moreover, we have
gcd(2m− 2e+ 2i+ 1,m− e+ i+ j) = gcd(2j − 1,m− e+ i+ j) < p
for 1 ⩽ j ⩽ i, from which it follows that
νp((m− e+ i+ 1)i) = 0.
By these comments and since
2i ⩽ e < p,
it follows from (7.7) (with n = 2m) that
νp
(
s2i
s0
)
= νp
(
(m− e+ 1)2i(ℓ− e+ 1)2i
(2m− 2e+ 2)2i
)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 199
= νp
(
(m− e+ i+ 1)i(ℓ− e+ 1)2i
2i(2m− 2e+ 3)(2m− 2e+ 5) · · · (2m− 2e+ 2i+ 1)
)
< 0.
However, this contradicts the fact that s2i/s0 is a non-zero integer. Hence we now conclude
that Θ must be finite.
The proof of Theorem 7.1 is complete.
ORCID iDs
Hajime Tanaka https://orcid.org/0000-0002-5958-0375
Yan Zhu https://orcid.org/0000-0001-7164-9198
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Appendix A Proof of Proposition 7.2
Our proof of Proposition 7.2 is a slight modification of (the first part of) that of [40, Satz I].
For a positive integer n, let
χn = 1 · 3 · 5 · · · (2n− 1) =
(2n)!
2nn!
.
Observe that the exponent νp(χn) of an odd prime p in χn is given by
νp(χn) =
⌊logp(2n)⌋∑
i=1
(⌊
2n
pi
⌋
−
⌊
n
pi
⌋)
=
⌊logp(2n)⌋∑
i=1
⌊
n
pi
+
1
2
⌋
, (A.1)
where we have used
⌊ξ⌋+
⌊
ξ +
1
2
⌋
= ⌊2ξ⌋ (ξ ∈ R).
Now, let (a, b) be a pair of positive integers with
b < c · aδ, (A.2)
which does not satisfy the desired property about a prime factor; in other words,
νp(χa+k)− νp(χa) ⩽ νp((b+ ⌊ϑk⌋)!)− νp(b!) if p > 2k+ 1. (A.3)
Our aim is to show that a is bounded in terms of ϑ, δ, c, and k, and hence so is b by (A.2),
from which it follows that there are only finitely many such pairs. (We will specify k0 =
k0(ϑ, δ) at the end of the proof.) To this end, we may assume for example that
a > k, c · aδ > k+ 1. (A.4)
Without loss of generality, we may also assume that
b > k, (A.5)
for otherwise the pair (a, k+ 1) would also satisfy (A.2) and (A.3).
Let
s =
χa+k
χkχa
· b!
(b+ ⌊ϑk⌋)!
.
Then from (A.3) it follows that
s ⩽
∏
3⩽p⩽2k+1
pνp(s), (A.6)
where the product in the RHS is over the odd primes p ⩽ 2k+ 1, and where
νp(s) = νp(χa+k)− νp(χk)− νp(χa) + νp(b!)− νp((b+ ⌊ϑk⌋)!).
By (A.1), for every odd prime p we have
νp(s) ⩽ νp(χa+k)− νp(χk)− νp(χa)
204 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
=
⌊logp(2a+2k)⌋∑
i=1
(⌊
a+ k
pi
+
1
2
⌋
−
⌊
k
pi
+
1
2
⌋
−
⌊
a
pi
+
1
2
⌋)
.
Note that ⌊
ξ + η +
1
2
⌋
−
⌊
ξ +
1
2
⌋
−
⌊
η +
1
2
⌋
∈ {−1, 0, 1} (ξ, η ∈ R).
Hence it follows that
νp(s) ⩽ logp(2a+ 2k) =
ln(2a+ 2k)
ln p
(A.7)
for every odd prime p. From (A.6) and (A.7) it follows that
ln s ⩽ (π(2k+ 1)− 1) ln(2a+ 2k), (A.8)
where π(n) denotes the number of primes at most n.
On the other hand, we have
s =
(2a+ 2k)!k!a!
(a+ k)!(2k)!(2a)!
· b!
(b+ ⌊ϑk⌋)!
.
Using Stirling’s formula
ln(n!) =
(
n+
1
2
)
ln n− n+ ln 2π
2
+ rn,
where
0 < rn <
1
12n
,
we obtain
ln s > (a+ k) ln(a+ k)− k ln k− a ln a+ ϑk− 2
+
(
b+
1
2
)
ln b−
(
b+ ϑk+
1
2
)
ln(b+ ϑk). (A.9)
Let
ã =
a
k
, b̃ =
b
k
.
Note that
ã, b̃ > 1, (A.10)
in view of (A.4) and (A.5). With this notation, we have
ln s > k ln ã+ (ã+ 1)k ln
(
1 +
1
ã
)
− ϑk ln k+ ϑk− 2
− ϑk ln b̃−
(
(b̃+ ϑ)k+
1
2
)
ln
(
1 +
ϑ
b̃
)
> k ln ã− ϑk ln k− 2− ϑk ln b̃−
(
ϑk+
1
2
)
ln
(
1 +
ϑ
b̃
)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 205
> (1− ϑδ)k ln ã− ϑk ln c− ϑδk ln k− 2−
(
ϑk+
1
2
)
ln(1 + ϑ), (A.11)
where the first inequality is a restatement of (A.9), the second follows from
0 < ln(1 + ξ) < ξ (ξ > 0),
and the last one follows from (A.2) and (A.10).
Concerning the prime-counting function π(n), it is known that [37, (3.6)]
π(n) < 1.25506
n
ln n
(n > 1).
By this, (A.8), and (A.10), we have
ln s ⩽ (π(2k+ 1)− 1)
(
ln ã+ ln 2k
(
1 +
1
ã
))
<
(
1.25506
2k+ 1
ln(2k+ 1)
− 1
)
(ln ã+ ln 4k). (A.12)
Combining (A.11) and (A.12), it follows that(
(1− ϑδ)k− 1.25506 2k+ 1
ln(2k+ 1)
+ 1
)
ln ã
< ϑk ln c+ ϑδk ln k+ 2 +
(
ϑk+
1
2
)
ln(1 + ϑ)
+
(
1.25506
2k+ 1
ln(2k+ 1)
− 1
)
ln 4k. (A.13)
If we set
k0 = k0(ϑ, δ) =
1
2
(
exp
(
2.51012
1− ϑδ
)
− 1
)
> 0
for example, then we have
(1− ϑδ)k− 1.25506 2k+ 1
ln(2k+ 1)
+ 1 ⩾
1 + ϑδ
2
> 0 (k ⩾ k0).
Hence, whenever k ⩾ k0, it follows from (A.13) that ln a = ln ã+ ln k is bounded in terms
of ϑ, δ, c, and k, from which and (A.2) it follows that there are only finitely many choices
for the pairs (a, b).
This completes the proof of Proposition 7.2.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.02 / 207–216
https://doi.org/10.26493/1855-3974.2593.1b7
(Also available at http://amc-journal.eu)
Two-distance transitive normal Cayley graphs*
Jun-Jie Huang , Yan-Quan Feng † , Jin-Xin Zhou
School of mathematics and statistics, Beijing Jiaotong University,
Beijing 100044, P. R. China
Received 3 April 2021, accepted 10 July 2021, published online 14 April 2022
Abstract
In this paper, we construct an infinite family of normal Cayley graphs, which are 2-
distance-transitive but neither distance-transitive nor 2-arc-transitive. This answers a ques-
tion proposed by Chen, Jin and Li in 2019.
Keywords: Cayley graph, 2-distance-transitive graph, simple group.
Math. Subj. Class. (2020): 05C25, 05E18, 20B25
1 Introduction
In this paper, all graphs are finite, simple, and undirected. For a graph Γ , let V (Γ ), E(Γ ),
A(Γ ) or Aut(Γ ) denote its vertex set, edge set, arc set and its full automorphism group, re-
spectively. The graph Γ is called G-vertex-transitive, G-edge-transitive or G-arc-transitive,
with G ≤ Aut(Γ ), if G is transitive on V (Γ ), E(Γ ) or A(Γ ) respectively, and G-semi-
symmetric, if Γ is G-edge-transitive but not G-vertex-transitive. It is easy to see that a
G-semisymmetric graph Γ must be bipartite such that G has two orbits, namely the two
parts of Γ , and the stabilizer Gu for any u ∈ V (Γ ) is transitive on the neighbourhood of u
in Γ . An s-arc of Γ is a sequence v0, v1, . . . , vs of s+1 vertices of Γ such that vi−1, vi are
adjacent for 1 ≤ i ≤ s and vi−1 ̸= vi+1 for 1 ≤ i ≤ s− 1. If Γ has at least one s-arc and
G ≤ Aut(Γ ) is transitive on the set of s-arcs of Γ , then Γ is called (G, s)-arc-transitive,
and Γ is said to be s-arc-transitive if it is (Aut(Γ ), s)-arc-transitive.
For two vertices u and v in V (Γ ), the distance d(u, v) between u and v in Γ is the
smallest length of paths between u and v, and the diameter diam(Γ ) of Γ is the maximum
distance occurring over all pairs of vertices. For i = 1, 2, . . . ,diam(Γ ), denote by Γi(u)
*The work was supported by the National Natural Science Foundation of China (11731002, 12011540376,
12011530455, 12071023, 12161141005) and the 111 Project of China (B16002).
†Corresponding author.
E-mail addresses: 20118006@bjtu.edu.cn (Jun-Jie Huang), yqfeng@bjtu.edu.cn (Yan-Quan Feng),
jxzhou@bjtu.edu.cn (Jin-Xin Zhou)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
208 Ars Math. Contemp. 22 (2022) #P2.02 / 207–216
the set of vertices at distance i with vertex u in Γ . A graph Γ is called distance transitive
if, for any vertices u, v, x, y with d(u, v) = d(x, y), there exists g ∈ Aut(Γ ) such that
(u, v)g = (x, y). The graph Γ is called (G, t)-distance-transitive with G ≤ Aut(Γ ) if,
for each 1 ≤ i ≤ t, the group G is transitive on the ordered pairs of form (u, v) with
d(u, v) = i, and Γ is said to be t-distance-transitive if it is (Aut(Γ ), t)-distance-transitive.
Distance-transitive graphs were first defined by Biggs and Smith in [2], and they showed
that there are only 12 trivalant distance-transitive graphs. Later, distance-transitive graphs
of valencies 3, 4, 5, 6 and 7 were classified in [2, 10, 14, 25], and a complete classification
of distance-transitive graphs with symmetric or alternating groups of automorphisms was
given by Liebeck, Praeger and Saxl [18]. The 2-distance-transitive but not 2-arc-transitive
graphs of valency at most 6 were classified in [4, 16], and the 2-distance-primitive graphs
(a vertex stabilizer of automorphism group is primitive on both the first step and the second
step neighbourhoods of the vertex) with prime valency were classified in [15]. By defini-
tion, a 2-arc-transitive graph is 2-distance-transitive, but a 2-distance-transitive graph may
not be 2-arc-transitive; an example is the Kneser graph KG6,2, see [16]. Furthermore, Corr,
Jin and Schneider [5] investigated properties of a connected (G, 2)-distance-transitive but
not (G, 2)-arc-transitive graph of girth 4, and they applied the properties to classify such
graphs with prime valency. For more information about 2-distance-transitive graphs, we
refer to [6, 7].
For a finite group G and a subset S ⊆ G \ {1} with S = S−1 := {s−1 | s ∈ S}, the
Cayley graph Cay(G,S) of the group G with respect to S is the graph with vertex set G
and with two vertices g and h adjacent if hg−1 ∈ S. For g ∈ G, let R(g) be the permutation
of G defined by x 7→ xg for all x ∈ G. Then R(G) := {R(g) | g ∈ G} is a regular group
of automorphisms of Cay(G,S). It is known that a graph Γ is a Cayley graph of G if and
only if Γ has a regular group of automorphisms on the vertex set which is isomorphic to G;
see [1, Lemma 16.3] and [24]. A Cayley graph Γ = Cay(G,S) is called normal if R(G)
is a normal subgroup of Aut(Γ ). The study of normal Cayley graphs was initiated by Xu
[27] and has been investigated under various additional conditions; see [8, 22].
There are many interesting examples of arc-transitive graphs and 2-arc-transitive graphs
constructed as normal Cayley graphs. However, the status for 2-distance-transitive graphs
is different. Recently, 2-distance-transitive circulants were classified in [3], where the fol-
lowing question was proposed:
Question 1.1 ([3, Question 1.2]). Is there a normal Cayley graph which is 2-distance-
transitive, but neither distance-transitive nor 2-arc-transitive?
In this paper, we answer the above question by constructing an infinite family of such
graphs, which are Cayley graphs of the extraspecial p-groups of exponent p of order p3.
Theorem 1.2. For an odd prime p, let G = ⟨a, b, c | ap = bp = cp = 1, [a, b] = c, [c, a] =
[c, b] = 1⟩ and S = {ai, bi | 1 ≤ i ≤ p − 1}. Then Cay(G,S) is a 2-distance-transitive
normal Cayley graph that is neither distance-transitive nor 2-arc-transitive.
A clique of a graph Γ is a maximal complete subgraph, and the clique graph Σ of Γ is
defined to have the set of all cliques of Γ as its vertex set with two cliques adjacent in Σ
if the two cliques have at least one common vertex. Applying Theorem 1.2, we can obtain
the following corollary.
Corollary 1.3. Under the notation given in Theorem 1.2, let Cos(G, ⟨a⟩, ⟨b⟩) be the graph
with vertex set {⟨a⟩g | g ∈ G} ∪ {⟨b⟩h | h ∈ G} and with edges all these coset pairs
J.-J. Huang et al.: Two-distance transitive normal Cayley graphs 209
{⟨a⟩g, ⟨b⟩h} having non-empty intersection in G. Then Cos(G, ⟨a⟩, ⟨b⟩) is the clique
graph of Cay(G,S), and Cay(G,S) is the line graph of Cos(G, ⟨a⟩, ⟨b⟩). Furthermore,
Cos(G, ⟨a⟩, ⟨b⟩) is 3-arc-transitive.
The graph Cos(G, ⟨a⟩, ⟨b⟩) was first constructed in [19] as a regular cover of Kp,p,
where it is said that Cos(G, ⟨a⟩, ⟨b⟩) is 2-arc-transitive in [19, Theorem 1.1], but not 3-arc-
transitive generally for all odd primes p in a remark after [19, Example 4.1]. However, this
is not true and Corollary 1.3 implies that Cos(G, ⟨a⟩, ⟨b⟩) is always 3-arc-transitive for each
odd prime p. In fact, Cos(G, ⟨a⟩, ⟨b⟩) is 3-arc-regular, that is, Aut(Cos(G, ⟨a⟩, ⟨b⟩)) is
regular on the set of 3-arcs of Cos(G, ⟨a⟩, ⟨b⟩). Some more information about the structure
and symmetry properties of Cos(G, ⟨a⟩, ⟨b⟩) are given in Lemma 3.2.
2 Preliminaries
In this section we list some preliminary results used in this paper. The first one is the
well-known orbit-stabilizer theorem (see [9, Theorem 1.4A]).
Proposition 2.1. Let G be a group with a transitive action on a set Ω and let α ∈ Ω. Then
|G| = |Ω||Gα|.
The well-known Burnside paqb theorem was given in [12, Theorem 3.3].
Proposition 2.2. Let p and q be primes and let a and b be positive integers. Then a group
of order paqb is soluble.
The next proposition is an important property of a non-abelian simple group acting
transitively on a set with cardinality a prime-power, whose proof depends on the finite
simple group classification, and we refer to [13, Corollary 2] or [26, Proposition 2.4].
Proposition 2.3. Let T be a nonabelian simple group acting transitively on a set Ω with
cardinality a p-power for a prime p. If p does not divide the order of a point-stabilizer of
T , then T acts 2-transitively on Ω .
Let Γ = Cay(G,S) be a Cayley graph of a group G with respect to S. Then R(G)
is a regular subgroup of Aut(Γ ), and Aut(G,S) := {α ∈ Aut(G) | Sα = S} is also a
subgroup of Aut(Γ ), which fixes 1. Furthermore, R(G) is normalized by Aut(G,S), and
hence we have a semiproduct R(G) ⋊ Aut(G,S), where R(g)α = R(gα) for any g ∈ G
and α ∈ Aut(G,S). Godsil [11] proved that the semiproduct R(G)⋊Aut(G,S) is in fact
the normalizer of R(G) in Aut(Γ ). By Xu [27], we have the following proposition.
Proposition 2.4. Let Γ = Cay(G,S) be a Cayley graph of a finite group G with respect
to S, and let A = Aut(Γ ). Then the following hold:
(1) NA(R(G)) = R(G)⋊Aut(G,S);
(2) Γ is a normal Cayley graph if and only if A1 = Aut(G,S), where A1 is the stabilizer
of 1 in A.
Let Γ be a G-vertex-transitive graph, and let N be a normal subgroup of G. The normal
quotient graph ΓN of Γ induced by N is defined to be the graph with vertex set the orbits
of N and with two orbits B,C adjacent if some vertex in B is adjacent to some vertex in
C in Γ . Furthermore, Γ is called a normal N -cover of ΓN if Γ and ΓN have the same
valency.
210 Ars Math. Contemp. 22 (2022) #P2.02 / 207–216
Proposition 2.5. Let Γ be a connected G-vertex-transitive graph and let N be a normal
subgroup of G. Suppose that either Γ is an N -cover of ΓN , or Γ is G-arc-transitive of
prime valency and N has at least three orbits on vertices. Then the following statements
hold:
(1) N is semiregular on V Γ and is the kernel of G acting V (ΓN ), so G/N ≤ Aut(ΓN );
(2) Γ is (G, s)-arc-transitive if and only if ΓN is (G/N, s)-arc-transitive;
(3) Gα ∼= (G/N)δ for any α ∈ V Γ and δ ∈ V (ΓN ).
Proposition 2.5 was given in many papers by replacing the condition that Γ is a normal
N -cover of ΓN by one of the following assumptions: (1) N has at least 3-orbits and G
is 2-arc-transitive (see [21, Theorem 4.1]); (2) N has at least 3-orbits, G is arc-transitive
and Γ has a prime valency (see [20, Theorem 2.5]); (3) N has at least 3-orbits and G is
locally primitive (see [17, Lemma 2.5]). The first step for these proofs is to show that for
any two vertices B,C ∈ V (ΓN ), the induced subgraph [B] of B in Γ has no edge and if
B and C are adjacent in ΓN then the induced subgraph [B ∪ C] in Γ is a matching, which
is equivalent to that Γ is a normal N -cover of ΓN . Then Proposition 2.5(1) - (3) follows
from these proofs.
3 Proof Theorem 1.2
For a positive integer n and a prime p, we use Zn and Zrp to denote the cyclic group of
order n and the elementary abelian group of order pr, respectively. In this section, we
always assume that p is an odd prime, and denote by Z∗p the multiplicative group of Zp
consisting of all non-zero numbers in Zp. Note that Z∗p ∼= Zp−1. Furthermore, we also set
the following assumptions in this section:
G = ⟨a, b, c | ap = bp = cp = 1, [a, b] = c, [c, a] = [c, b] = 1⟩,
S = {ai, bi | 1 ≤ i ≤ p− 1},
Γ = Cay(G,S), A = Aut(Γ ), N = NA(R(G)) = R(G)⋊Aut(G,S), and Z∗p = ⟨t⟩.
By Proposition 2.4, NA(R(G)) = R(G) ⋊ Aut(G,S), and R(g)δ = R(gδ) for any
R(g) ∈ R(G) and δ ∈ Aut(G,S). Since G = ⟨S⟩, Γ is a connected Cayley graph of
valency 2(p− 1). Let
α : a 7−→ at, b 7−→ b, c 7−→ ct;
β : a 7−→ a, b 7−→ bt, c 7−→ ct;
γ : a 7−→ b, b 7−→ a, c 7−→ c−1.
It is easy to check that at, b, ct satisfy the same relations as a, b, c in G, that is,
[at, b] = ct, [ct, at] = [ct, b] = 1. By the von Dyck’s Theorem (see [23, 2.2.1]), α in-
duces an epimorphism from G to ⟨at, b, ct⟩, which must be an automorphism of G because
⟨at, b, ct⟩ = G. Similarly, β and γ are also automorphisms of G.
Lemma 3.1. Aut(G,S) = ⟨α, β, γ⟩ ∼= (Zp−1 × Zp−1) ⋊ Z2, and Γ is N -arc-transitive.
Furthermore, N has no normal subgroup of order p2.
J.-J. Huang et al.: Two-distance transitive normal Cayley graphs 211
Proof. Since Z∗p = ⟨t⟩, it is easy to check that αp−1 = βp−1 = γ2 = 1, αβ = βα and
αγ = β. Thus ⟨α, β, γ⟩ ∼= (Zp−1 × Zp−1)⋊ Z2. Clearly, α, β, γ ∈ Aut(G,S). To prove
Aut(G,S) = ⟨α, β, γ⟩ ∼= (Zp−1 × Zp−1) ⋊ Z2, it suffices to show that |Aut(G,S)| ≤
2(p− 1)2.
Clearly, ⟨α, β, γ⟩ is transitive on S, and hence Γ is N -arc-transitive. Since G = ⟨S⟩,
Aut(G,S) is faithful on S. By Proposition 2.1, |Aut(G,S)| = |S||Aut(G,S)a|, where
Aut(G,S)a is the stabilizer of a in Aut(G,S). Note that Aut(G,S)a fixes ai for each
1 ≤ i ≤ p− 1. Again by Proposition 2.1, |Aut(G,S)a| ≤ (p− 1)|Aut(G,S)a,b|, where
Aut(G,S)a,b is the subgroup of Aut(G,S) fixing a and b. Since G = ⟨a, b⟩, we obtain
Aut(G,S)a,b = 1, and then |Aut(G,S)| ≤ 2(p− 1)2, as required.
Let H ≤ N be a subgroup of order p2. Since R(G) is the unique normal Sylow p-
subgroup of N = R(G) ⋊ Aut(G,S), we have H ≤ R(G), and since |R(G) : H| = p,
we have H ⊴ R(G). Note that the center C := Z(R(G)) = ⟨R(c)⟩ and C ∩ H ̸= 1.
Thus, C ∩ H = C as |C| = p, implying C ≤ H . Since H/C is a subgroup of order p,
and R(G)/C = ⟨R(a)C⟩ × ⟨R(b)C⟩ ∼= Z2p, we have H/C = ⟨R(b)C⟩ or ⟨R(a)R(b)iC⟩
for some 0 ≤ i ≤ p − 1. It follows that H = ⟨R(b)⟩ × C or ⟨R(abi)⟩ × C for some
0 ≤ i ≤ p− 1.
Suppose H ⊴N . Since C is characteristic in R(G) and R(G) ⊴N , we have C ⊴N .
Recall that R(a)γ = R(aγ) = R(b). Then (⟨R(a)⟩×C)γ = ⟨R(b)⟩×C. This implies that
both ⟨R(a)⟩×C and ⟨R(b)⟩×C are not normal in N . Thus, H = ⟨R(abi)⟩×C for some
1 ≤ i ≤ p − 1. Since H ⊴ N , we have Hβ = H , that is, ⟨R(abti)⟩ × C = Hβ = H =
⟨R(abi)⟩ × C. It follows that ⟨R(abti)⟩ = ⟨R(abi)⟩ and then R(abti) = R(abi), which
further implies bti = bi. This gives rise to p | i(t− 1), and since (i, p) = 1, we have t = 1,
contradicting that Z∗p = ⟨t⟩ ∼= Zp−1. Thus, N has no normal subgroup of order p2.
For a positive integer n, np denotes the largest p-power diving n. By Lemma 3.1,
Γ = Cay(G,S) is N -arc-transitive.
Lemma 3.2. The clique graph Σ of Γ is a connected p-valent bipartite graph of order 2p2,
A has a faithful natural action on Σ , and Σ is R(G)-semisymmetric and N -arc-transitive.
Furthermore, |A|p = p3.
Proof. Recall that G = ⟨a, b, c | ap = bp = cp = 1, [a, b] = c, [c, a] = [c, b] = 1⟩ and
S = {ai, bi | 1 ≤ i ≤ p − 1}. Then Γ = Cay(G,S) has exactly two cliques passing
through 1, that is, the induced subgraphs of ⟨a⟩ and ⟨b⟩ in Γ . Since R(G) ≤ Aut(Γ )
is transitive on vertex set, each clique of Γ is an induced subgraph of the coset ⟨a⟩x or
⟨b⟩x for some x ∈ G. Thus, we may view the vertex set of Σ as {⟨a⟩x, ⟨b⟩x | x ∈ G}
with two cosets adjacent in Σ if they have non-empty intersection. It is easy to see that
⟨a⟩x ∩ ⟨b⟩y ̸= ∅ if and only if |⟨a⟩x ∩ ⟨b⟩y| = 1, and any two distinct cosets, either in
{⟨a⟩x | x ∈ G} or in {⟨b⟩x | x ∈ G}, have empty intersection. Furthermore, ⟨a⟩ has
non-empty intersection with exactly p cosets, that is, ⟨b⟩ai for 0 ≤ i ≤ p− 1. Thus, Σ is a
p-valent bipartite graph of order 2p2. The connectedness of Σ follows from that of Γ .
Clearly, A has a natural action on Σ . Let K be the kernel of A on Σ . Then K fixes
each coset of ⟨a⟩x and ⟨b⟩x for all x ∈ G. Since ⟨a⟩x ∩ ⟨b⟩x = {x}, K fixes x and hence
K = 1. Thus, A is faithful on Σ and we may let A ≤ Aut(Σ ).
Note that R(G) is not transitive on {⟨a⟩x, ⟨b⟩x | x ∈ G}, but transitive on
{⟨a⟩x | x ∈ G} and {⟨b⟩x | x ∈ G}. Furthermore, R(⟨a⟩) fixes ⟨a⟩ and is transitive
on {⟨b⟩ai | 0 ≤ i ≤ p− 1}, the neighbourhood of ⟨a⟩ in Σ , and similarly, R(⟨b⟩) fixes ⟨b⟩
212 Ars Math. Contemp. 22 (2022) #P2.02 / 207–216
and is transitive on the neighbourhood {⟨a⟩bi | 0 ≤ i ≤ p−1} of ⟨b⟩ in Σ . It follows that Σ
is R(G)-semisymmetric. Recall that N = R(G)⋊Aut(G,S) and Aut(G,S) = ⟨α, β, γ⟩.
Since aγ = b and bγ = a, γ interchanges {⟨a⟩x | x ∈ G} and {⟨b⟩x | x ∈ G}. This yields
that Σ is R(G)⋊ ⟨γ⟩-arc-transitive and hence N -arc-transitive.
Since Σ is a connected graph with prime valency p, we have p2 ∤ |Aut(Σ )u| for
any u ∈ V (Σ ), and in particular, p2 ∤ |Au|. Note that p | |Au|. By Proposition 2.1,
|A| = |Σ ||Au| = 2p2|Au|. This implies that |A|p = p3.
Lemma 3.3. A = Aut(Γ ) = R(G)⋊Aut(G,S).
Proof. By Lemma 3.2, |A|p = p3, and since |V (Γ )| = p3 and A is vertex-transitive on
V (Γ ), the vertex stabilizer A1 is a p′-group, that is, p ∤ |A1|. To prove the lemma, by
Proposition 2.4 we only need to show that R(G)⊴A, and since R(G) is a Sylow p-subgroup
of A, it suffices to show that A has a normal Sylow p-subgroup.
Let M be a minimal normal subgroup of A. Then M = T1×T2 · · ·×Td, where Ti ∼= T
for each 1 ≤ i ≤ d with a simple group T . Since |V (Γ )| = p3, each orbit of M has length
a p-power and hence each orbit of Ti has length a p-power. It follows that p | |T |. Assume
that |T |p = pℓ. Then |M |p = pdℓ and dℓ = 1, 2 or 3 as |A|p = p3.
We process the proof by considering the two cases: M is insoluble or soluble.
Case 1: M is insoluble.
In this case, T is a non-abelian simple group. We prove that this case cannot happen by
deriving contradictions. Recall that dℓ = 1, 2 or 3.
Assume that dℓ = 1. Then |M |p = p. By Lemma 3.2, M ⊴ A ≤ Aut(Σ ), and since
|V (Σ )| = 2p2, M has at least three orbits. Since Σ has valency p, Proposition 2.5 implies
that M is semiregular on V (Σ ) and hence |M | | 2p2. By Proposition 2.2, M is soluble, a
contradiction.
Assume that dℓ = 2. Since R(G) is a Sylow p-subgroup of A and M ⊴ A, R(G) ∩M
is a Sylow p-subgroup of M and hence |R(G) ∩M | = |M |p = p2. Since R(G)⊴N and
M ⊴ A, M ∩R(G) is a normal subgroup of order p2 in N , contradicting to Lemma 3.1.
Assume that dℓ = 3. Then (d, ℓ) = (1, 3) or (3, 1). Since |M |p = p3 = |A|p, we
deduce R(G) ≤ M and hence M is transitive on Γ .
For (d, ℓ) = (1, 3), M is a non-abelian simple group. Since M1 ≤ A1 is a p′-group,
Proposition 2.3 implies that M is 2-transitive on Γ , forcing that Γ is the complete graph of
order p3, a contradiction.
For (d, ℓ) = (3, 1), we have M = T1 × T2 × T3. Then |M |p = p3, and since M ⊴ A,
we derive R(G) ≤ M . By Lemma 3.2 M ≤ Aut(Σ ), and Σ is R(G)-semisymmetric.
Since M has no subgroup of index 2, M fixes the two parts of Σ setwise, and hence Σ is
M -semisymmetric. Noting that γ interchanges the two parts of Σ , we have that Σ is M⟨γ⟩-
arc-transitive. Since γ is an involution, under conjugacy it fixes Ti for some 1 ≤ i ≤ 3, say
T1. Then T1 ⊴ ⟨M,γ⟩ and by Proposition 2.5, T1 is semiregular on Σ . This gives rise to
|T1| | 2p2, contrary to the simplicity of T1.
Case 2: M is soluble.
Since p | |M |, we have M = Zdp with 1 ≤ d ≤ 3. If d = 3 then A has a normal Sylow
p-subgroup, as required. If d = 2 then M ≤ R(G) ≤ N and N has a normal subgroup
of order p2, contrary to Lemma 3.1. Thus, we may let d = 1, and since M ≤ R(G) and
R(G) has a unique normal subgroup of order p that is the center of R(G), we derive that
M = ⟨R(c)⟩.
J.-J. Huang et al.: Two-distance transitive normal Cayley graphs 213
Now it is easy to see that the quotient graph ΓM = Cay(G/M,S/M) with S/M =
{aiM, biM | 1 ≤ i ≤ p − 1}. Note that G/M = ⟨aM⟩ × ⟨bM⟩ ∼= Z2p. Then ΓM is a
connected Cayley graph of order p2 with valency 2(p − 1), so Γ is a normal M -cover of
ΓM . By Proposition 2.5, we may let A/M ≤ Aut(ΓM ) and ΓM is A/M -arc-transitive.
Let H/M be a minimal normal subgroup of A/M . Then H ⊴A and H/M = L1/M ×
· · · × Lr/M , where Li ⊴H and Li/M (1 ≤ i ≤ r) are isomorphic simple groups. Since
|ΓM | = p2, we infer p | |H/M | and similarly, p | |Li/M |. Let |Li/M |p = ps. Then
|H/M |p = prs, and since |A/M |p = p2, we obtain that sr = 1 or 2.
We finish the proof by considering the two subcases: H/M is insoluble or soluble.
Subcase 2.1: H/M is insoluble.
In this subcase, Li/M are isomorphic non-abelain simple groups. We prove this sub-
case cannot happen by deriving contradictions. Recall that sr = 1 or 2.
Let sr = 1. Then |H/M |p = p, and therefore |H|p = p2. Since H ⊴ A, H ∩ R(G)
is a Sylow p-subgroup of H , implying |H ∩ R(G)| = p2, and then R(G) ⊴N yields that
H ∩R(G) is a normal subgroup of order p2 in N , contrary to Lemma 3.1.
Let rs = 2. Then |H/M |p = p2 and |H|p = p3. This yields R(G) ≤ H and H is
transitive on Γ , so H/M is transitive on V (ΓM ). Note that (r, s) = (1, 2) or (2, 1).
For (r, s) = (1, 2), H/M is a nonabelian simple group. By Propostion 2.5, (H/M)u
for u ∈ V (ΓM ) is a p′-group because H1 ≤ A1 is a p′-group, and by Proposition 2.3, H/M
is 2-transitive on V (ΓM ), forcing that ΓM is a complete group of order p2, a contradiction.
For (r, s) = (2, 1), H/M ∼= L1/M × L2/M , where L1/M and L2/M are isomorphic
nonabelain simple groups and |Li/M |p = p. It follows that |H|p = p3 and |Li|p = p2 for
1 ≤ i ≤ 2. Since H ⊴ A, we derive R(G) ≤ H . Note that H has no subgroup of index 2.
Since Σ is bipartite, it is H-semisymmetric. Let ∆1 and ∆2 be the two parts of Σ . Then
|∆1| = |∆2| = p2, and H is transitive on both ∆1 and ∆2.
Suppose (L1)u = 1 for some u ∈ V (Σ ) = ∆1∪∆2. By Proposition 2.1, |L1| = |uL1 |,
and since L1 ⊴ H and |∆1| = |∆2| = p2, we derive |L1| = p or p2, contrary to the
insolubleness of L1. Thus (L1)u ̸= 1. Since Σ has prime valency p, Hu is primitive on the
neighbourhood Σ (u) of u in Σ , and since (L1)u ⊴Hu, (L1)u is transitive on Σ (u), which
implies that |(L1)u|p = p. Since |L1|p = p2, each orbit of L1 on ∆1 or ∆2 has length p.
Let x ∈ ∆1 and y ∈ ∆2 be adjacent in Σ , and let ∆11 and ∆21 be the orbits of L1
containing x and y, respectively. Then |∆11| = |∆21| = p. Since (L1)x is transitive on
Σ (x), x is adjacent to each vertex in ∆21, and therefore, each vertex in ∆11 is adjacent
to each vertex in ∆21, that is, the induced subgroup [∆11 ∪∆21] is the complete bipartite
graph Kp,p. It follows that Σ ∼= pKp,p, contrary to the connectedness of Σ .
Subcase 2.2: H/M is soluble.
In this case, |H| = p2 or p3. Recall that H ⊴ A. If |H| = p2 then H ≤ R(G) and
N has normal subgroup of order p2, contradicts Lemma 3.1. Thus, |H| = p3 and A has a
normal Sylow p-subgroup, as required. This completes the proof.
Now we are ready to finish the proof.
Proof of Theorem 1.2. By Lemmas 3.1 and 3.3, Γ is a arc-transitive normal Cayley graph.
In particular, Γ is 1-distance transitive. Since S = {ai, bi | 1 ≤ i ≤ p− 1}, Γ has girth 3,
so it is not 2-arc-transitive.
214 Ars Math. Contemp. 22 (2022) #P2.02 / 207–216
Recall that G = ⟨a, b, c | ap = bp = cp = 1, [a, b] = c, [c, a] = [c, b] = 1⟩. Clearly,
Γ1(1) = S = {ai, bi | 1 ≤ i ≤ p− 1},
Γ2(1) = {bjai, ajbi | 1 ≤ i, j ≤ p− 1}.
Note that Aut(G,S) = ⟨α, β, γ | αp−1 = βp−1 = γ2 = 1, αβ = α, αγ = β⟩, where
aα = at, bα = b, cα = ct, aβ = a, bβ = bt, cβ = ct, aγ = b, bγ = a and cγ = c−1.
Then (ba)α
iβj = bt
i
at
j
, and since Z∗p = ⟨t⟩, we obtain that ⟨α, β⟩ is transitive on the set
{bjai | 1 ≤ i, j ≤ p − 1}. Similarly, ⟨α, β⟩ is transitive on {ajbi | 1 ≤ i, j ≤ p − 1}.
Furthermore, γ interchanges the two sets {bjai | 1 ≤ i, j ≤ p− 1} and {ajbi | 1 ≤ i, j ≤
p−1}. It follows that Aut(G,S) is transitive on Γ2(1) and hence Γ is 2-distance transitive.
Noting that ab = bac, we have that b−1ab = ac ∈ Γ3(1) and aba = ba2c ∈ Γ3(1).
Also it is easy to see that (ac)Aut(G,S) = (ac)⟨α,β,γ⟩ = {aicj , bicj | 1 ≤ i, j ≤ p−1}. Now
it is easy to see that ba2c ̸∈ (ac)Aut(G,S), and since A1 = Aut(G,S) by Proposition 2.4,
Γ is not distance-transitive.
Proof of Corollary 1.3. Recall that Σ is the clique graph of Γ . By the first paragraph
in the proof of Lemma 3.2 and the definition of Cos(G, ⟨a⟩, ⟨b⟩) in Corollary 1.3, we
have Σ = Cos(G, ⟨a⟩, ⟨b⟩). Again by Lemma 3.2, Σ is R(G)-semisymmetric, and since
|E(Σ )| = (2p2 · p)/2 = p3 = |R(G)|, R(G) is regular on the edge set E(Σ ) of Σ . Thus,
the line graph of Σ is a Cayley graph on G.
For a given edge {⟨a⟩x, ⟨b⟩y} ∈ E(Σ ), we have |⟨a⟩x ∩ ⟨b⟩y| = 1, and then we may
identify this edge with the unique element in ⟨a⟩x∩⟨b⟩y. Note that Σ has valency 2(p−1).
Then the edge 1 = ⟨a⟩ ∩ ⟨b⟩ in Σ is exactly incident to all edges in S = {ai, bi | 1 ≤ i ≤
p− 1}, because {ai} = ⟨a⟩∩ ⟨b⟩ai and {bi} = ⟨b⟩∩ ⟨a⟩bi. It follows that Γ = Cay(G,S)
is exactly the line graph of Σ .
If α ∈ Aut(Σ ) fixes each edge in Σ then α fixes all vertices of Σ , that is, Aut(Σ ) acts
faithfully on Γ . Thus, we may view Aut(Σ ) as a subgroup of Aut(Γ ). By Lemmas 3.2
and 3.3, we have Aut(Γ ) = Aut(Σ ) = R(G)⋊Aut(G,S).
Recall that Aut(G,S) = ⟨α, β, γ⟩ and Σ is arc-transitive. Since aβ = a, bβ = bt and
cβ = ct, where Z∗p = ⟨t⟩, ⟨β⟩ fixes the arc (⟨a⟩, ⟨b⟩) in Σ and is transitive on the vertex set
{⟨a⟩bi | 1 ≤ i ≤ p − 1}, where {⟨a⟩} ∪ {⟨a⟩bi | 1 ≤ i ≤ p − 1} is the neighbourhood of
⟨b⟩ in Σ . Thus, Σ is 2-arc-transitive. Since aα = at, bα = b and cα = ct, ⟨α⟩ fixes the
2-arc (⟨a⟩, ⟨b⟩, ⟨a⟩b) and is transitive on the vertex set {⟨b⟩aib | 1 ≤ i ≤ p − 1}, where
{⟨b⟩} ∪ {⟨b⟩aib | 1 ≤ i ≤ p − 1} is the neighbourhood of ⟨a⟩b in Σ . It follows that Σ is
3-arc-transitive. It is easy to see that the number of 3-arcs in Σ equals to |A| = 2p3(p−1)2,
A is regular on the set of 3-arcs of Σ .
ORCID iDs
Jun-Jie Huang https://orcid.org/0000-0002-9250-0672
Yan-Quan Feng https://orcid.org/0000-0003-3214-0609
Jin-Xin Zhou https://orcid.org/0000-0002-8353-896X
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.03 / 217–229
https://doi.org/10.26493/1855-3974.2501.4c4
(Also available at http://amc-journal.eu)
LDPC codes from cubic semisymmetric graphs*
Dean Crnković , Sanja Rukavina , Marina Šimac †
Department of Mathematics, University of Rijeka,
Radmile Matejčić 2, 51000 Rijeka, Croatia
Received 10 December 2020, accepted 16 July 2021, published online 14 April 2022
Abstract
In this paper we study LDPC codes having cubic semisymmetric graphs as their Tanner
graphs. We discuss the structure of the smallest absorbing sets of these LDPC codes.
Further, we give the expression for the variance of the syndrome weight of the constructed
codes, and present computational and simulation results.
Keywords: LDPC code, cubic graph, semisymmetric graph.
Math. Subj. Class. (2020): 94B05, 05C99
1 Introduction and preliminaries
Throughout this paper we assume graphs to be finite, simple and connected. For the con-
cepts and notation related to the graph theory and coding theory, we refer the reader to [10]
and [15], respectively.
In this paper we use cubic semisymmetric graphs for the construction of LDPC codes.
A graph is called a 3-regular graph, i.e. a cubic graph, if every vertex of the graph has
the degree equal to three. A graph is edge-transitive (vertex-transitive) if its automorphism
group acts transitively on the set of edges (set of vertices). A regular graph is semisymmet-
ric if it is edge-transitive, but not vertex-transitive. It has been proved that every semisym-
metric graph is necessarily bipartite with two parts of equal size (see [14]).
Semisymmetric graphs were first studied by Folkman in 1967 (see [12]). He proposed
a construction of semisymmetric graphs and constructed the smallest semisymmetric graph
with 20 vertices and 40 edges (the Folkman graph). Furthermore, it has been proved that
there are no semisymmetric graphs with 2p or 2p2 vertices for a prime number p.
*This work has been fully supported by Croatian Science Foundation under the project 6732.
†Corresponding author.
E-mail addresses: deanc@math.uniri.hr (Dean Crnković), sanjar@math.uniri.hr (Sanja Rukavina),
msimac@math.uniri.hr (Marina Šimac)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
218 Ars Math. Contemp. 22 (2022) #P2.03 / 217–229
A cubic semisymmetric graph is a 3-regular graph which is semisymmetric. A con-
struction of cubic semisymmetric graphs and the (non)existence of graphs with a certain
number of vertices have been a subject of many studies. For example, in [20], the exis-
tence of the unique cubic semisymmetric graph with 2p3 vertices for a prime number p,
the Gray graph of order 54, was proved. In [11], the condition for the existence of cubic
semisymmetric graphs with 6p3 vertices was given, and a construction of such graphs was
described. The classification of cubic semisymmetric graphs with at most 768 vertices was
given in [4]. All of the listed graphs have girth at least eight.
The dual code C⊥ of an [n, k] linear code C is an [n, n− k] code defined by
C⊥ =
{
x ∈ Fnp | x · y = 0, ∀y ∈ C
}
,
where · is the standard inner product. A generator matrix of the code C⊥ is called a parity-
check matrix of C.
A binary low-density parity-check (LDPC) code is a binary linear code defined by a
sparse parity-check matrix H . That is to say, H contains a very small number of nonzero
entries. An LDPC code is (wc, wr)-regular if the weight of each column is equal to wc,
and the weight of each row is equal to wr.
LDPC codes can be presented using Tanner graphs, which were introduced by Tanner
in [26]. The Tanner graph of an LDPC code is a bipartite graph that consists of two sets
of vertices; bit nodes that correspond to codeword bits and check nodes that correspond to
parity-check equations. An edge connects a bit node to a check node if that bit is included
in the corresponding parity-check equation. If an LDPC code is (wc, wr)-regular, the cor-
responding Tanner graph is a biregular bipartite graph in which vertices are of degree wc
or wr.
The decoding performance of an LDPC code depends on the structure of the corre-
sponding Tanner graph; the existence of short cycles in the Tanner graph of a code es-
tablishes a correlation between iterations in the process of decoding, and therefore, has a
negative impact on the bit error rate (BER) performance of the code. The shorter the cy-
cles are, the more significant the effect is. Furthermore, the iterative decoding performance
of an LDPC code is related with the existence of certain undesirable substructures of the
corresponding Tanner graph. For an AWGN channel, substructures that are called trapping
sets, determine error floor performance of an LDPC code. It has been proved that absorbing
sets, as a special type of trapping sets, have an important role in the error floor (see [25]).
Various combinatorial structures, including graphs, were used for a construction of
LDPC codes without cycles of length four (see, e.g., [6, 16, 17, 23]). In [7], the authors in-
vestigated a family of LDPC codes constructed by taking bipartite cubic symmetric graphs
as the Tanner graphs. In this paper, we construct LDPC codes from cubic semisymmetric
graphs and study the smallest absorbing sets in the corresponding Tanner graphs.
The paper is organized as follows. In Section 2, the construction of the family of LDPC
codes using cubic semisymmetric graphs is presented, some properties of the obtained
codes are analyzed and the results regarding the code parameters are given. Furthermore,
the expression for the variance of the syndrome weight of the constructed LDPC codes is
presented. In Section 3, the structure of the smallest absorbing sets is studied. Sections 4
and 5 contain computational and simulation results.
D. Crnković et al.: LDPC codes from cubic semisymmetric graphs 219
2 LDPC codes constructed from cubic semisymmetric graphs
Let G be a connected cubic semisymmetric graph with 2n vertices, and denote by A its
adjacency matrix. Since every semisymmetric graph is bipartite with two parts of equal
size, its adjacency matrix can be written as follows
A =
[
0 H
HT 0
]
, (2.1)
where H is an n× n matrix.
Taking the matrix H as a parity-check matrix, one can construct a (3, 3)-regular LDPC
code CH(G) of length n. The dimension of that code is equal to n − rank2(H), where
rank2(H) =
1
2 rank2(A). Furthermore, the density of the parity-check matrix H is equal
to 3n . For the constructed code CH(G), the cubic semisymmetric graph G is its Tanner
graph.
From the fact that semisymmetric graphs are edge-transitive, but not vertex-transitive, it
follows that HT determines another LDPC code CHT (G). The code CHT (G) is a
(3, 3)-regular LDPC code of length n, and its dimension is equal to n − rank2(H) as
well.
Let H and HT be n × n parity-check matrices of the codes CH(G) and CHT (G), re-
spectively. For the code CH(G), the bit node graph Γb is defined in the following way:
vertices of the graph correspond to codeword bits, and two vertices are adjacent if and only
if the corresponding bits are included in the same parity-check equation. In other words,
two vertices of the graph Γb are adjacent if and only if the corresponding bit nodes of the
Tanner graph of the code CH(G) have a common neighbour. Similarly, the vertices of the
check node graph Γc correspond to parity-check equations of the code, and two vertices
are adjacent if and only if the corresponding parity-check equations have a bit in common.
That is to say, two vertices of the graph Γc are adjacent if and only if the corresponding
check nodes of the Tanner graph of the code CH(G) have a common neighbour. Note that
the check node graph Γc of the code CH(G) is the bit node graph of the code CHT (G).
Theorem 2.1. Let G be a connected cubic semisymmetric graph with girth at least six and
let H be the parity-check matrix of the code CH(G). Then the corresponding bit node graph
Γb and check node graph Γc are 6-regular.
Proof. Let v be a bit node of the Tanner graph G. The degree of the node v is equal to
three, and each of its neighbours is adjacent to another two bit nodes. Using the fact that G
does not have cycles of length four, it follows that v has a common neighbour with exactly
six other bit nodes. In other words, the degree of a vertex of the graph Γb is equal to six,
i.e., the graph Γb is 6-regular. In the same way it can be concluded that the graph Γc is also
6-regular.
Theorem 2.2. Let G be a connected cubic semisymmetric graph with 2n vertices and girth
at least six. Further, let H be the parity-check matrix of the code CH(G) and let Γb and Γc
be the corresponding bit node graph and check node graph, respectively. Matrices Tb and
Tc are square (0, 1)-matrices of order n satisfying Tb = HTH − 3I and Tc = HHT − 3I
if and only if Tb and Tc are the adjacency matrices of the graphs Γb and Γc, respectively.
220 Ars Math. Contemp. 22 (2022) #P2.03 / 217–229
Proof. Let us consider the n × n matrix HTH = [hi,j ]. The degree of a bit node of the
Tanner graph G of the code CH(G) is equal to three, hence hi,i = 3, i ∈ {1, . . . , n}. An
element hi,j , i ̸= j, of the matrix H is equal to one or zero depending on whether the
corresponding nodes of the graph Γb are adjacent or not. Accordingly, Tb = HTH − 3I ,
where Tb is the adjacency matrix of the graph Γb.
Conversely, let Tb = [ti,j ] be an n × n (0, 1)-matrix with the property that
Tb = H
TH − 3I . HTH is a symmetric matrix and, consequently, Tb is also a sym-
metric matrix such that ti,i = 0, i ∈ {1, . . . , n}. The girth of the Tanner graph G is greater
than four, so hi,j , i ̸= j, is equal to zero or one, and represents the number of common
neighbours of the corresponding bit nodes of the Tanner graph G of the code CH(G). It
follows that Tb is the adjacency matrix of the graph Γb.
An analog statement for the matrix Tc can be formed similarly by observing check
nodes of the Tanner graph of the code CH(G).
A clique of a graph G is a complete subgraph of the graph G. The clique number of the
graph G, denoted by ω(G), is the number of vertices in a clique of the largest size in G, i.e.
the order of a complete subgraph of G of maximum possible size for G. In the sequel, the
clique number of the bit node graph Γb and the check node graph Γc will be examined.
Lemma 2.3. Let G be a connected cubic semisymmetric graph. Further, let CH(G) be
the corresponding LDPC code and let Γb and Γc be its bit node and check node graph,
respectively. The clique numbers of the graphs Γb and Γc are at least three.
Proof. Each check node of the Tanner graph G is a common neighbour of every pair of
its three adjacent bit nodes. Thus, each check node determines the complete graph K3
as a subgraph of the bit node graph Γb. Similarly, each bit node of the Tanner graph
determines the complete graph K3 as a subgraph of the check node graph Γc. Hence,
ω(Γb), ω(Γc) ≥ 3.
Lemma 2.4. Let G be a connected cubic semisymmetric graph with girth greater than six.
Further, let CH(G) be the corresponding LDPC code and let Γb and Γc be its bit node and
check node graph, respectively. Then the complete graph K4 is not a subgraph of Γb or
Γc.
Proof. Suppose that K4 is a subgraph of the graph Γb. Let the bit nodes u1, u2, u3, u4 be
the vertices of K4. We have the following two possibilities:
(a) One of the check nodes (say v1) in the corresponding subgraph of the Tanner graph G
has degree three. Let u1, u2 and u3 be the bit nodes adjacent with v1. Furthermore,
let the check node v2 be a common neighbour of u1 and u4. Since u2 and u4 are
adjacent in Γb, they have a common neighbour v3 in G. Then u1v1u2v3u4v2u1 is a
cycle of length six, which is impossible since the girth of the graph G is greater than
six.
(b) The check nodes in the corresponding subgraph of the Tanner graph G have degrees
at most two. Let the check node vi be a common neighbour of the bit nodes u1 and
ui+1, i = 1, 2, 3. Since u2 and u4 are adjacent in Γb, they have a common neighbour
v4 in G. Then u1v1u2v4u4v3u1 is a cycle of length six, which contradicts the fact
that the girth of the graph G is greater than six.
D. Crnković et al.: LDPC codes from cubic semisymmetric graphs 221
Analog arguments yield that K4 is not a subgraph of Γc.
The following theorem is a direct consequence of Lemmas 2.3 and 2.4.
Theorem 2.5. Let G be a connected cubic semisymmetric graph with girth greater than
six. Further, let CH(G) be the corresponding LDPC code and let Γb and Γc be its bit node
and check node graph, respectively. Then ω(Γb) = ω(Γc) = 3.
In the sequel, we discuss the minimum distance of the codes CH(G) and CHT (G). The
following results from [24] will be used.
Theorem 2.6 ([24, Theorem 3.1]). Let C be a binary linear code with a parity-check matrix
H . Then there exists a codeword in C with weight w if and only if there are w columns in
H whose vector sum is a zero vector.
Theorem 2.7 ([24, Theorem 3.2]). Let C be a binary linear code with a parity-check matrix
H . Then the minimum distance of the code C is equal to the smallest number of columns in
H whose vector sum is a zero vector.
The column weight of parity check matrices H and HT of codes CH(G) and CHT (G) is
equal to three, and according to Theorem 2.6, the codes are even. Therefore, the minimum
distance of the codes is an even number.
Theorem 2.8. Let G be a connected cubic semisymmetric graph with girth greater than six.
Let d(CH(G)) and d(CTH(G)) be the minimum distances of the codes CH(G) and CHT (G),
respectively. Then d(CH(G)) ≥ 6 and d(CTH(G)) ≥ 6.
Proof. The column weight of the parity-check matrix H of the code CH(G) is equal to
three, and since the graph G does not have cycles of length four, it follows that the minimum
distance of the code is at least four (see [13]). Assume that the minimum distance of the
code is equal to four. As a consequence of Theorem 2.7, four columns of the parity-check
matrix whose sum equals zero exist. Therefore, a set S in the graph G, which consists of
four bit nodes such that each pair of the vertices has a different common neighbour in G,
exists. Moreover, the set S determines the complete graph K4 as a subgraph of the bit node
graph Γb. Using Theorem 2.5, we conclude that the minimum distance of the code is at
least six.
Observing check nodes of the Tanner graph of the code CH(G), and the check node
graph Γc, one can prove the statement for the minumum distance of the code CHT (G).
In [7, Theorem 1], the minimum distance of an LDPC code constructed from a bipartite
cubic symmetric graph is expressed using the second largest eigenvalue of the adjacency
matrix of that graph. In a similar way, using the result given in Theorem 2.8, one can prove
the following theorem.
Theorem 2.9. Let G be a connected cubic semisymmetric graph with 2n vertices and girth
greater than six. Let λ2 be the second largest eigenvalue of its adjacency matrix A. Let
d(CH(G)) and d(CTH(G)) be the minimum distances of the codes CH(G) and CHT (G), re-
spectively. Then the following inequalities hold
d ≥

2
5n, λ2 ≤ 2,
2
9n, 2 < λ2 ≤
√
6,
6,
√
6 < λ2 < 3,
222 Ars Math. Contemp. 22 (2022) #P2.03 / 217–229
where d ∈ {d(CH(G)), d(CTH(G))}.
Remark 2.10. The results given above refer to the LDPC codes constructed from con-
nected cubic semisymmetric graphs with girth greater than six. According to the classifi-
cation of cubic semisymmetric graphs with at most 768 vertices (see [4]), all such graphs
have girth at least eight. Consequently, all of the associated LDPC codes have properties
stated above.
Theorem 2.11. Let G be a connected cubic semisymmetric graph with 2n vertices. Then
the dimension of the codes CH(G) and CHT (G) is at most n− 2α(Γb) + 1, where α(Γb) is
the independence number of the bit node graph Γb.
Proof. The 2-rank of the parity-check matrix of a binary code determines its dimension.
The 2-rank of the matrix H is equal to the 2-rank of the matrix HT and, therefore, it is
sufficent to observe the matrix H and the corresponding code CH(G). A maximal indepen-
dent set of Γb determines α(Γb) linearly independent columns of the parity check matrix
H . These columns have the property that no two columns have an entry equal to one at
the same position. Due to the fact that Γb is a 6-regular graph, there are 6α(Γb) ones at
different positions within the columns. Therefore, adding any other α(Γb)− 1 columns of
the matrix, a set of 2α(Γb)− 1 linearly independent columns of the parity check matrix is
defined. Hence, 2-rank of the matrix H is at least 2α(Γb)− 1.
As a consequence, the dimension of the code is at most n − 2α(Γb) + 1, where n
is the length of the code, i.e. the number of vertices of the graph Γb, and α(Γb) is the
independence number of the graph Γ.
2.1 The variance of syndrome weight
To predict a decoding efficiency one can use a channel state information (CSI) (e.g. the
crossover probability, a signal-to-noise ratio), which has an important role for communi-
cation systems. The estimation (performed prior to decoding) of the crossover probability
based on the probability of syndrome weight was proposed in [18] and [27].
The expression for the variance of the syndrome weight of the LDPC codes constructed
from bipartite cubic symmetric graphs is given in [7]. In a similar way, one can obtain the
expression for the variance of the syndrome weight of the LDPC codes constructed from
cubic semisymmetric graphs which is given by
V ar(w) =
n
2
(7f6(ρ)− 6f4(ρ)) ,
where the function ft is defined by ft(ρ) =
1−(1−2ρ)t
2 (see [22]).
3 Absorbing sets
Let G = G(C) be the Tanner graph of an LDPC code C which is determined by an m× n
parity check matrix H . A (κ, τ) trapping set is a set T , that consists of κ bit nodes, having
the property that the induced subgraph G[T ] has exactly τ check nodes of odd degree. The
most harmful trapping sets are those with small sizes and small ratios τκ . If the Tanner
graph of an LDPC code does not have trapping sets with size smaller than the minimum
distance of the code, then the error floor of the code is dominated by the minimum distance
(see [9]). Let T be a trapping set. If every bit node in G[T ] is connected with fewer check
nodes of odd degree than check nodes of even degree, then T is called an absorbing set.
D. Crnković et al.: LDPC codes from cubic semisymmetric graphs 223
Let A be a (κ, τ)− trapping set in the Tanner graph of an (3, wr) LDPC code. Using
simple counting it can be seen that τ is an even number if κ is even, and an odd number if
κ is odd.
The results in the sequel refer to the LDPC codes for which the corresponding Tanner
graphs have girth at least six. We examine the existence of the smallest absorbing sets in
the Tanner graphs of the LDPC codes constructed from the cubic semisymmetric graphs.
Theorem 3.1. Let the Tanner graph of the LDPC code CH(G) be a connected cubic
semisymmetric graph G with girth at least six. Then there is no absorbing set of size
smaller than three in the graph G.
Proof. The proof follows directly from the definition of an absorbing set and the fact that
the Tanner graph of the code has no cycles of length four.
Theorem 3.2. Let G be a connected cubic semisymmetric graph with girth greater than
six, which is the Tanner graph of the LDPC codes CH(G) and CHT (G). The Tanner graph
G has no absorbing set of size three.
Proof. Let A be a (3, 3)-absorbing set, which is the only possible structure of an absorbing
set of size three in the Tanner graph of the codes (see Figure 1). The proof follows directly
from the fact that the absorbing set defines a cycle of length six in the Tanner graph.
Figure 1: The only possible structure of an absorbing set of size three in the Tanner graph
of the LDPC codes CH(G). and CHT (G).
Theorem 3.3. Let G be a connected cubic semisymmetric graph with girth greater than
six, which is the Tanner graph of the LDPC codes CH(G) and CHT (G). The only possible
structure for an absorbing set of size four is (4, 4)-absorbing set.
Proof. Since the size of an absorbing set is an even number, and according to the previous
observations, the possible structures for absorbing sets of size four in the Tanner graph of
the codes are (4, 0), (4, 2) and (4, 4) absorbing sets (see Figure 2(a), (b) and (c), respec-
tively). The proof follows directly from the fact that (4, 0) and (4, 2) absorbing sets define
the complete graph K4 as a subgraph of the graph Γb.
224 Ars Math. Contemp. 22 (2022) #P2.03 / 217–229
(a)
(b)
(c)
Figure 2: The possible structures of an absorbing set of size four in the Tanner graph of the
LDPC codes CH(G) and CHT (G).
4 Computational results
Within this section the parameters of the LDPC codes obtained from cubic semisymmetric
graphs are presented. For the construction of the cubic semisymmetric graphs we have
employed the method presented in [1]. The parameters of the constructed codes can be
seen in Table 1. The parameter v denotes the number of vertices of the corresponding
cubic semisymmetric graph.
v LDPC1 LDPC2
54 [27, 8, 6] [27, 8, 8]∗
112 [56, 12, 14] [56,12,16]
120 [60, 14, 8] [60,14,12]
144 [72, 16, 12]∗ [72, 16, 14]∗
216 [108, 16, 24] [108, 16, 32]
240 [120, 22, 16] [120, 22, 24]
294 [147, 26, 14] [147, 26, 26]
336 [168, 24, 14] [168, 24, 42]
378 [189, 11, 42] [189, 11, 56]
384 [192, 35, 16] [192, 35, 18]
400 [200,24,32] [200,24,60]
432 [216, 24, 48] [216, 24, 60]
v LDPC1 LDPC2
448 [224, 33, 32] [224,33,32]
486 [243, 2, 162]∗ [243, 2, 162]∗
546 [273, 5, 130] [273, 5, 130]
576 [288, 32, 48] [288, 32, 56]
672 [336, 47, 14] [336, 47, 42]
702 [351, 8, 78] [351, 8, 104]∗
720 [360,10,120] [360,10,120]
784 [392, 12, 98] [392,12,112]
798 [399, 5, 190] [399, 5, 190]
864 [432, 32, 96] [432, 32, 108]
882 [441, 44, 42] [441, 44, 78]
896 [448, 48, 84] [448,48,100]
Table 1: The parameters of LDPC codes constructed from cubic semisymmetric graphs
with less than 1000 vertices (using the method presented in [1]).
D. Crnković et al.: LDPC codes from cubic semisymmetric graphs 225
The Tanner graphs of the constructed codes have girth at least eight. The codes CH(G)
and CHT (G) are isomorphic in the case when the number of vertices of the cubic semisym-
metric graph G is 486, 546, 720 or 798.
Remark 4.1. Lately, much interest has been devoted to LCD codes, which have an impor-
tant application in cryptography, in protection against side-channel and fault attacks (see
[2]). Self-orthogonal codes can be used to construct quantum error-correcting codes, which
can protect quantum information in quantum computations and quantum communications
(see [3]).
The LDPC codes marked in bold are self-orthogonal codes, and those labeled with ∗ in
Table 1 are LCD codes.
Remark 4.2. Codes CH(G) and CHT (G) constructed from a cubic semisymmetric graph
(CSSG) have the same length and dimension, and, in general, different minimum distance.
Thus, the construction gives diversity in code parameters for the same graph, which is
not the case for LDPC codes which are constructed in [7] using cubic symmetric graphs
(CSGs).
According to the classification of CSSGs with at most 768 vertices (see [4]), all the
graphs have girth at least eight, while according to [5] many CSGs have girth equal to six.
Moreover, semisymmetric graphs form a wider family than symmetric graphs.
Furthermore, we have compared the parameters of the LDPC codes constructed from
CSSGs to the parameters of the LDPC codes constructed from CSGs. The results are shown
in Table 2. It can be concluded that, for the same code length, the LDPC codes from CSSGs
achieve higher code rate than those constructed using CSGs. When n = 27, the code rate
is four times greater.
n Rate (CSSG) Rate (CSG)
27 0.296 0.074
56 0.214 {0.107, 0.143}
60 0.233 {0.067, 0.083}
72 0.222 {0.083, 0.111}
Table 2: Rates od LDPC codes constructed from cubic symmetric and semisymmetric
graphs with the same length.
5 Simulation results
In this section, we present simulation results of the LDPC codes derived from the cubic
semisymmetric graphs, over the additive white gaussian noise (AWGN) channel. We have
compared the codes with randomly generated LDPC codes of the same length and dimen-
sion and a parity-check matrix with a column weight equal to three. For randomly gener-
ated codes we have used the software for LDPC codes available on [21], which employs the
construction from [8, 19]. The codes are decoded with the sum-product decoding algorithm
and the maximum number of iteration is set to 50. Figures 3 - 6 show the performance of
the codes.
226 Ars Math. Contemp. 22 (2022) #P2.03 / 217–229
Remark 5.1. The LDPC codes that we are aware of were not adequate for the comparison
with the LDPC codes obtained in this paper because of the different parameters of the
codes. Thus, we have used the best known random construction for LDPC codes. It has
been proved in [8] that the construction leads to LDPC codes with performance close to the
Shannon limit. Moreover, the best results were obtained in the case of the smallest possible
column weight.
Figure 3: BER performance of the [56, 12, 16] LDPC code derived from the Ljubljana
graph.
Figure 4: BER performance of the [147, 26, 26] LDPC code derived from the cubic
semisymmetric graph with 294 vertices.
D. Crnković et al.: LDPC codes from cubic semisymmetric graphs 227
Figure 5: BER performance of the [288, 32, 56] LDPC code derived from the cubic
semisymmetric graph with 576 vertices.
Figure 6: BER performance of the [448, 48, 100] LDPC code derived from the cubic
semisymmetric graph with 896 vertices.
The obtained simulation results indicate better BER performance of the codes con-
structed from the cubic semisymmetric graphs than randomly generated LDPC codes.
ORCID iDs
Dean Crnković https://orcid.org/0000-0002-3299-7859
Sanja Rukavina https://orcid.org/0000-0003-3365-7925
Marina Šimac https://orcid.org/0000-0001-9291-3365
228 Ars Math. Contemp. 22 (2022) #P2.03 / 217–229
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.04 / 231–248
https://doi.org/10.26493/1855-3974.2522.eb3
(Also available at http://amc-journal.eu)
Paired domination stability in graphs
Aleksandra Gorzkowska
AGH University, Department of Discrete Mathematics,
al. Mickiewicza 30, 30-059 Krakow, Poland
Michael A. Henning
Department of Mathematics and Applied Mathematics, University of Johannesburg,
Auckland Park, 2006 South Africa
Monika Pilśniak
AGH University, Department of Discrete Mathematics,
al. Mickiewicza 30, 30-059 Krakow, Poland
Elżbieta Tumidajewicz *
AGH University, Department of Discrete Mathematics,
al. Mickiewicza 30, 30-059 Krakow, Poland, and
Department of Mathematics and Applied Mathematics, University of Johannesburg,
Auckland Park, 2006 South Africa
Received 29 December 2020, accepted 16 July 2021, published online 27 May 2022
Abstract
A set S of vertices in a graph G is a paired dominating set if every vertex of G is
adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not
necessarily as an induced subgraph). The paired domination number, γpr(G), of G is the
minimum cardinality of a paired dominating set of G. A set of vertices whose removal from
G produces a graph without isolated vertices is called a non-isolating set. The minimum
cardinality of a non-isolating set of vertices whose removal decreases the paired domination
number is the γ−pr-stability of G, denoted st
−
γpr(G). The paired domination stability of G
is the minimum cardinality of a non-isolating set of vertices in G whose removal changes
the paired domination number. We establish properties of paired domination stability in
graphs. We prove that if G is a connected graph with γpr(G) ≥ 4, then st−γpr(G) ≤ 2∆(G)
where ∆(G) is the maximum degree in G, and we characterize the infinite family of trees
that achieve equality in this upper bound.
Keywords: Paired domination, paired domination stability.
*Corresponding author.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
232 Ars Math. Contemp. 22 (2022) #P2.04 / 231–248
Math. Subj. Class. (2020): 05C69
1 Introduction
In 1983 Bauer, Harary, Nieminen and Suffel [3] introduced and studied the concept of
domination stability in graphs. Stability for other domination type parameters has been
studied in the literature. For example, total domination stability, 2-rainbow domination
stability, exponential domination stability, Roman domination stability are studied in [1, 2,
12, 15, 16], among other papers. In this paper we study the paired version of domination
stability.
Let G = (V,E) be a graph with vertex set V = V (G) and edge set E = E(G). Two
vertices u and v are neighbors if they are adjacent, that is, if uv ∈ E. A dominating set of G
is a set D of vertices such that every vertex in V (G)\D has a neighbor in D. The minimum
cardinality of a dominating set is the domination number, γ(G), of G. Domination is well
studied in the literature. A recent book on domination in graphs can be found in [10]. A
small sample of papers on domination critical graphs can be found in [3, 4, 5, 6, 9, 17, 18].
Adopting the notation coined by Bauer et al. [3], the γ−-stability (γ+-stability, resp.) of
G, denoted by γ−(G) (γ+(G), resp.), is the minimum number of vertices whose removal
decreases (increases, resp.) the domination number. The minimum number of vertices
whose removal decreases or increases the domination number is the domination stability,
stγ(G), of G, and so stγ(G) = min{γ−(G), γ+(G)}.
We refer to a graph without isolated vertices as an isolate-free graph. Unless otherwise
stated, let G be an isolate-free graph. A total dominating set, abbreviated TD-set, of G
is a set D of vertices of G such that every vertex, including vertices in the set D, has
a neighbor in D. The minimum cardinality of a TD-set of G is the total domination number,
γt(G), of G. We call a TD-set of G of cardinality γt(G) a γt-set of G. A vertex v is totally
dominated by a set D in G if the vertex v has a neighbor in D. We refer the reader to
the book [14] for fundamental concepts on total domination in graphs. Total domination
critical graphs are studied, for example, in [7, 13]. The total version of domination stability
was first studied by Henning and Krzywkowski [12].
A paired dominating set, abbreviated PD-set, of an isolate-free graph G is a dominating
set S of G with the additional property that the subgraph G[S] induced by S contains
a perfect matching M (not necessarily induced). With respect to the matching M , two
vertices joined by an edge of M are paired and are called partners in S. The paired
domination number, γpr(G), of G is the minimum cardinality of a PD-set of G. We call
a PD-set of G of cardinality γpr(G) a γpr-set of G. We note that the paired domination
number γpr(G) is an even integer. For a recent survey on paired domination in graphs, we
refer the reader to the book chapter [8].
Every PD-set is a TD-set, implying that γt(G) ≤ γpr(G). A non-isolating set of ver-
tices in G is a set S ⊆ V such that the graph G − S is isolate-free, where G − S is the
graph obtained from G by removing S and all edges incident with vertices in S. Let NI(G)
denote the set of all non-isolating sets of vertices of G.
Adopting the standard notation for domination stability given in [3, 12], the γ−pr-stability
E-mail addresses: agorzkow@agh.edu.pl (Aleksandra Gorzkowska), mahenning@uj.ac.za (Michael A.
Henning), pilsniak@agh.edu.pl (Monika Pilśniak), etumid@agh.edu.pl (Elżbieta Tumidajewicz)
A. Gorzkowska et al.: Paired domination stability in graphs 233
(resp., γ+pr-stability) of G, denoted by st
−
γpr(G) (resp., st
+
γpr(G)) is the minimum cardinality
of a set in NI(G) whose removal decreases (increases, resp.) the paired domination number.
Thus,
st−γpr(G) = minS∈NI(G)
{ |S| : γpr(G− S) < γpr(G)}
and
st+γpr(G) = minS∈NI(G)
{ |S| : γpr(G− S) > γpr(G)}.
If there is no set in NI(G) whose removal increases the paired domination number,
then we define st+γpr(G) = ∞. For example, st
−
γpr(P5) = 1 while st
+
γpr(P5) = ∞.
The paired domination stability, stγpr(G), of G is the minimum cardinality of a set in
NI(G) whose removal increases or decreases the paired domination number. Thus,
stγpr(G) = min
S∈NI(G)
{ |S| : γpr(G− S) ̸= γpr(G)} = min{st−γpr(G), st
+
γpr(G)}.
Let G be a graph and let S ∈ NI(G). If γpr(G − S) < γpr(G) and |S| = st−γpr(G), then
we call S a st−γpr -set of G. If γpr(G − S) > γpr(G) and |S| = st
+
γpr(G), then we call S
a st+γpr -set of G. If γpr(G − S) ̸= γpr(G) and |S| = stγpr(G), then we call S a stγpr -set
of G.
Defining the null graph K0, which has no vertices, as a graph, we have the following
results due to Bauer et al. [3] and Rad et al. [15] for the γ−-stability of a graph.
Theorem 1.1 ([3, 15]). If G is an isolate-free graph of order n, then the following holds.
(a) stγ(G) ≤ δ(G) + 1.
(b) If G ≇ Kn, then stγ(G) ≤ n− 1.
Considering the null graph, the paired domination stability of a non-trivial graph is
always defined. If G is a graph of order n and γpr(G) = 2, then st−γpr(G) = n since
removing all vertices from the graph G produces the null graph with paired domination
number zero.
For notation and graph theory terminology we generally follow [14]. In particular, for
r, s ≥ 1, a double star S(r, s) is the tree with exactly two vertices that are not leaves, one
of which has r leaf-neighbors and the other s leaf-neighbors. A rooted tree is a tree T in
which we specify one vertex r called the root. For each vertex v of T different from r,
its parent is the neighbor of v on the unique (r, v)-path, while every other neighbor of v
is a child of v in T . If w is a vertex of T different from v and the (unique) (r, w)-path
contains v, then w is a descendant of v in T . We note that every child of v is a descendant
of v. The diameter diam(G) of G is the maximum distance among all pairs of vertices of
G. A diametral path in G is a shortest path between two vertices in G of length equal to
diam(G). For an integer k ≥ 1, [k] = {1, . . . , k}.
2 Main results
Our first aim is to show that the paired domination stability of a graph can be very different
from its total domination stability studied in [12].
Theorem 2.1. For k ≥ 1 an arbitrary integer, the following holds.
234 Ars Math. Contemp. 22 (2022) #P2.04 / 231–248
(a) There exist connected graphs G such that st−γpr(G)− st
−
γt(G) = k.
(b) There exist connected graphs H such that st−γt(H)− st
−
γpr(H) = k.
Our second aim is to establish properties of paired domination stability in graphs.
Thereafter, we establish upper bounds on the paired domination stability and the γ−pr-
stability of a graph. For this purpose, we shall need the following family of trees defined by
Henning and Krzywkowski [12]. For integers k ≥ 2 and ∆ ≥ 2, the authors in [12] define
Tk,∆ as the “graph obtained from the disjoint union of k double stars S(∆− 1,∆− 1) by
adding k − 1 edges between the leaves of these double stars so that the resulting graph is
a tree with maximum degree ∆." Let Fk,∆ be the family of all such trees Tk,∆, and let
F∆ =
⋃
k≥2
Fk,∆.
The following result establishes an upper bound on the γ−pr-stability of a tree, and char-
acterizes the trees with maximum possible γ−pr-stability.
Theorem 2.2. If T is a tree with maximum degree ∆ satisfying γpr(T ) ≥ 4, then the
following hold.
(a) st−γpr(T ) ≤ 2∆, with equality if and only if T ∈ F∆.
(b) stγpr(T ) ≤ 2∆− 1, and this bound is sharp for all ∆ ≥ 2.
For general graphs, we establish the following upper bound on the γ−pr-stability in terms
of the maximum degree of the graph.
Theorem 2.3. If G is a connected graph with γpr(G) ≥ 4, then st−γpr(G) ≤ 2∆(G), and
this bound is sharp.
As an immediate consequence of Theorem 2.3, we have the following upper bound on
the paired domination stability of a graph.
Corollary 2.4. If G is a connected graph with γpr(G) ≥ 4, then stγpr(G) ≤ 2∆(G).
3 Paired stability versus domination and total stability
In this section, we show that paired domination stability and the domination stability of
a graph can be very different. By Theorem 1.1, for every nontrivial graph G, we have
stγ(G) ≤ δ(G) + 1. In particular, stγ(T ) ≤ 2 for every nontrivial tree T . This is in
contrast to the paired domination stability, where for any given ∆ ≥ 2, we show that there
exist a family of trees T with maximum degree ∆ satisfying stγpr(T ) = 2∆− 1.
For ∆ = 2, the authors in [12] define H∆ as the family of all paths of order at least 7
and congruent to 3 modulo 4, that is, H∆ = {Pn | n ≡ 3 (mod 4) and n ≥ 7}. For
integers ∆ ≥ 3 and ∆ ≥ k ≥ 2, they define Hk,∆ as the graph “obtained from the disjoint
union of k double stars S(∆ − 1,∆ − 1) by selecting one leaf from each double star and
identifying these k leaves into one new vertex" and they define the family
H∆ =
⋃
k≥2
Hk,∆.
We determine next the paired domination stability of a tree in the family H∆.
A. Gorzkowska et al.: Paired domination stability in graphs 235
Proposition 3.1. For ∆ ≥ 3, if T ∈ H∆, then stγpr(T ) = 2∆− 1.
Proof. For integers ∆ ≥ k ≥ 2 where ∆ ≥ 3, consider a tree T ∈ Hk,∆. By definition
of the family Hk,∆, the tree T is constructed from the disjoint union of k double stars
S1, . . . , Sk, each isomorphic to S(∆ − 1,∆ − 1), by selecting one leaf from each double
star and identifying these k chosen leaves into one new vertex, which we call vc. Let xi
and yi be the two central vertices of the double star Si for i ∈ [k], where xi is adjacent to
vc in T . Let D = ∪ki=1{xi, yi}. Since ∆ ≥ 3, every vertex in D is a support vertex of
T , implying that every PD-set in T contains the set D and therefore γpr(T ) ≥ |D| = 2k.
Since the set D is a PD-set of T (with the vertices xi and yi paired for all i ∈ [k]), we have
γpr(T ) ≤ |D| = 2k. Consequently, γpr(T ) = 2k and D is the unique γpr-set of T .
Let S be a stγpr -set of T . Thus, S is a set in NI(T ) with |S| = stγpr(T ) satisfying
γpr(T − S) ̸= γpr(T ) = 2k. We show that |S| ≥ 2∆ − 1. Suppose, to the contrary, that
|S| ≤ 2∆− 2. If the set S contains both xi and yi for some i ∈ [k], then since S is a non-
isolating set of T every leaf neighbor of xi and yi is also in S, implying that |S| ≥ 2∆− 1,
a contradiction. Hence, the set S contains at most one of xi and yi for every i ∈ [k]. Let
D∗ be a γpr-set of T − S, and so |D∗| ≠ 2k.
Suppose that vc ∈ S. In this case, if |S| = 1, then the paired domination numbers of
T and T − S are the same, a contradiction. Hence, |S| ≥ 2. If neither xi nor yi belong
to S for some i ∈ [k], then by the minimality of the non-isolating set S, no vertex of Ti
different from vc belongs to S, and so |D∗ ∩ V (Ti)| = 2. If S contains yi but not xi for
some i ∈ [k], then every leaf neighbor of yi is in S and by the minimality of the set S, no
leaf neighbor of xi belongs to S, and so |D∗ ∩ V (Ti)| = 2. Analogously, if S contains xi
but not yi for some i ∈ [k], then |D∗ ∩ V (Ti)| = 2. This is true for all i ∈ [k], implying
that |D∗| =
∑k
i=1 |D∗ ∩ V (Ti)| = 2k, a contradiction. Hence, vc /∈ S.
As observed earlier, the set S contains at most one of xi and yi for every i ∈ [k]. If
yi ∈ S and yj ∈ S for some i, j ∈ [k] where i ̸= j, then |S| ≥ 2∆, a contradiction. If
yi ∈ S and xj ∈ S for some i, j ∈ [k] where i ̸= j, then |S| ≥ 2∆ − 1, a contradiction.
If xi ∈ S and xj ∈ S for some i, j ∈ [k] where i ̸= j, then |S| ≥ 2∆− 2. In this case, by
the minimality of S we have S = (N [xi] ∪ N [xj ]) \ {vc, yi, yj} and |S| = 2∆ − 2. But
then T − S consists of three components, namely two stars isomorphic to K1,∆−1 and one
component belonging to the family T ∈ Hk−2,∆ with paired domination number 2(k− 2).
Thus, γpr(T − S) = 2+ 2+ 2(k− 2) = 2k, a contradiction. Therefore, stγpr(T ) = |S| ≥
2∆− 1, as claimed.
Conversely, if we take S = N(x1) ∪ N(y1) \ {vc}, then S ∈ NI(T ) and T − S ∈
Hk−1,∆. Thus, γpr(T − S) = 2(k − 1) < γpr(T ), and so stγpr(T ) ≤ st−γpr(T ) ≤ |S| =
2∆− 1. Consequently, stγpr(T ) = st−γpr(T ) = 2∆− 1.
As observed earlier, stγ(T ) ≤ 2 for every nontrivial tree T . By Proposition 3.1, paired
domination stability therefore differs significantly from domination stability. We show next
that the paired domination stability and the total domination stability of a graph can also be
very different.
Proposition 3.2. For k ≥ 1 an integer, there exist trees T such that st−γpr(T )−st
−
γt(T ) = k.
Proof. Let k ≥ 1 be a given integer, and let T = Tk be obtained from a path P5 given
by v1v2v3v4v5 by attaching k leaf neighbors to each of v1, v2 and v3 (see Figure 1). We
236 Ars Math. Contemp. 22 (2022) #P2.04 / 231–248
note that {v1, v2, v3, v4} is the unique γt-set of T and the unique γpr-set of T . In partic-
ular, γt(T ) = γpr(T ) = 4. If S = {v5}, then the set S is a non-isolating set of T and
γt(T − S) = |{v1, v2, v3}| = 3 < γt(T ), implying that st−γt(T ) = 1.
We show next that st−γpr(T ) = k + 1. Let S be a non-isolating set of T such that
γpr(T − S) < γpr(T ). We show that |S| ≥ k + 1. Suppose, to the contrary, that |S| ≤ k.
Let D be a γpr-set of T − S, and so |D| = γpr(T − S) = 2. Let Li denote the set of leaf
neighbors of vi for i ∈ [4]. If vi ∈ S for some i ∈ [3], then S contains all k leaf neighbors
of vi, and so |S| ≥ k + 1, a contradiction. Hence, S ∩ {v1, v2, v3} = ∅. If {v1, v3} ⊂ D,
then |D| ≥ 4, a contradiction. If v1 /∈ D, then L1 ⊆ S, implying that S = L1 and
|S| = k. However in this case, {v2, v3, v4} ⊂ D. If v3 /∈ D, then L3 ⊆ S, implying that
S = L3 and |S| = k. However in this case, {v1, v2, v4} ⊂ D. In both cases, |D| ≥ 4,
a contradiction. Therefore, |S| ≥ k + 1, implying that st−γpr(T ) ≥ k + 1. Conversely, if
S = L1 ∪ L4, then S is a non-isolating set of T such that γpr(T − S) = |{v2, v3}| <
γpr(T ), implying that st−γpr(T ) ≤ |S| = k + 1. Consequently, st
−
γpr(T ) = k + 1. Thus,
st−γpr(T )− st
−
γt(T ) = k.
   

v1 v2 v3 v4
v5

  . . .
k

  . . .
k

  . . .
k
Figure 1: A tree from the family Tk in the proof of Proposition 3.2.
Proposition 3.3. For k ≥ 1 an integer, there exist trees T such that st−γt(T )−st
−
γpr(T ) = k.
Proof. Let k ≥ 1 be a given integer, and let ℓ ≥ 2k + 1 be an integer. For i ∈ [k], let
Qi be obtained from a path vi1vi2vi3vi4vi5 by attaching ℓ leaf neighbors to each of vi3 , vi4
and vi5 , and let Li3 , Li4 and Li5 be the resulting sets of leaf neighbors of vi3 , vi4 and vi5 ,
respectively. Let Q be obtained from a path v1v2v3 by attaching ℓ leaf neighbors to each of
v1 and v2, and attaching k leaf neighbors to v3. Let Li be the resulting set of leaf neighbors
of vi for i ∈ [3]. Let T be obtained from the disjoint union of the paths Q,Q1, . . . , Qk
by adding the k edges v3vi1 for i ∈ [k]. Let A be the set of support vertices of T , and so
|A| = 3(k + 1).
Every TD-set of T contains all its support vertices, implying that γt(T ) ≥ |A|. Since
the set A is a TD-set of T , we have γt(T ) ≤ |A|. Consequently, γt(T ) = |A| = 3(k + 1).
Every PD-set of T contains the set A and at least one additional vertex from each path
Qi that is a neighbor of vi3 or vi5 for i ∈ [k], and at least one additional vertex that is
a neighbor of v1 or v3 since the vertices of every PD-set are paired, implying that γpr(T ) =
|A|+ k + 1 = 4(k + 1).
Let S be a non-isolating set of T such that γpr(T −S) < γpr(T ). If |S| < k, then every
support vertex of T remains a support vertex of T −S, implying that γpr(T −S) ≥ γpr(T ),
a contradiction. Hence, |S| ≥ k. Conversely, if S∗ = L3, then the set A\{v3} of all support
vertices of T − S∗, together with the vertices vi2 for i ∈ [k], form a PD-set of T − S∗,
implying that γpr(T − S∗) ≤ 4k + 2 < 4k + 4 = γpr(T ). Hence, st−γpr(T ) ≤ |S
∗| = k.
Consequently, st−γpr(T ) = k.
A. Gorzkowska et al.: Paired domination stability in graphs 237
We show next that st−γt(T ) = 2k. Let A
′ = A \ {v3}, and so |A′| = |A| − 1 =
3k + 2. Let S be a non-isolating set of T such that γt(T − S) < γt(T ). We show that
|S| ≥ 2k. Suppose, to the contrary, that |S| ≤ 2k − 1. Let D be a γt-set of T − S, and
so |D| = γt(T − S) ≤ 3k + 2. Since |S| < 2k < ℓ and each vertex in A′ has ℓ leaf
neighbors in T , we note that every vertex of A′ is a support vertex of T − S, implying that
A′ ⊆ D, and so 3k + 2 ≥ |D| ≥ |A′| = 3k + 2, implying that D = A′. In particular,
v3 /∈ D, implying that all k leaf neighbors of v3 belong to S; that is, L3 ⊆ S. If vi1 /∈ S
for some i ∈ [k], then in order to totally dominate the vertex vi1 , the vertex vi2 ∈ D,
contradicting our earlier observation that D = A′. Hence, vi1 ∈ S for all i ∈ [k], and so
|S| ≥ |L3|+k = 2k, a contradiction. Therefore, our original supposition that |S| ≤ 2k−1
is incorrect, implying that |S| ≥ 2k and st−γpr(T ) ≥ 2k. Conversely, if S
∗ consists of
all 2k neighbors of v3 different from v2 in T , then S∗ is a non-isolating set of T such
that γt(T − S∗) = |A′| < γt(T ), implying that st−γt(T ) ≤ |S
∗| = 2k. Consequently,
st−γt(T ) = 2k. Thus, st
−
γt(T )− st
−
γpr(T ) = k.
  v1 v2 v3


...k
 
. . .
l

 
. . .
l

 
. . .
k
   
 
. . .
l

 
. . .
l

 
. . .
l
   
 
. . .
l

 
. . .
l

 
. . .
l
Figure 2: A tree from the family T in the proof of Proposition 3.3.
Theorem 2.1 follows from Propositions 3.2 and 3.3. As further examples, we remark
that if P is the Petersen graph, then γt(P ) = 4 and γpr(P ) = 6. Further, if v is an arbitrary
vertex of P , then γt(P − v) = 4, and so st−γt(P ) ≥ 2. Moreover, if S consists of two
non-adjacent vertices of P , then γt(P − S) = 3, and so st−γt(P ) ≤ 2. Consequently,
st−γt(P ) = 2. However if v is an arbitrary vertex of P , then γpr(P − v) = 4, implying that
st−γpr(P ) = 1. Moreover, let Gk be a graph obtained from the Petersen graph by replacing
every vertex by a copy of a complete graph Kk for some k ≥ 1, and adding all edges
between two resulting complete graphs that correspond to two vertices of Gk (see Fig. 3).
The resulting graph Gk is a (4k−1)-regular, 3k-connected graph that satisfies γt(Gk) = 4
and st−γt(Gk) = 2k, and γpr(Gk) = 6 and st
−
γpr(Gk) = k. This yields the following result.
Proposition 3.4. For k ≥ 1 an integer, there exists (4k− 1)-regular, 3k-connected graphs
G such that st−γt(G)− st
−
γpr(G) = k.
4 Properties of paired domination stability
In this section, we present properties of paired domination stability in graphs. We begin
with the following property of paired domination in graphs.
238 Ars Math. Contemp. 22 (2022) #P2.04 / 231–248
Kk Kk
KkKk
Kk
Kk Kk
KkKk
Kk
Figure 3: A graph Gk obtained from the Petersen graph by replacing every vertex by Kk.
Proposition 4.1. Every connected isolate-free graph G contains a spanning tree T such
that γpr(T ) = γpr(G).
Proof. Since adding edges to a graph cannot increases its paired domination number, if
T is an isolate-free spanning subgraph of a graph G, then γpr(G) ≤ γpr(T ). Let D be
a γpr-set of G, and so D is a PD-set of G and |D| = γpr(G). Let M be a perfect matching
in the subgraph G[D] induced by D. Let T ′ be a spanning subgraph of G that consists
of the edges in M and for each vertex v outside D, an edge of G that joins v to exactly
one vertex of the dominating set D. If the resulting spanning subgraph T ′ is a tree, then
we let T = T ′. Otherwise, if the resulting spanning subgraph T ′ is a forest with ℓ ≥ 2
components, then we add ℓ − 1 edges from the edge set of the graph G between these
components, avoiding cycles, to construct a tree, which we call T . Since D is a PD-set
in the resulting tree T , we note that γpr(T ) ≤ |D| = γpr(G). Since T is an isolate-free
spanning subgraph of G, we have γpr(T ) ≥ γpr(G). Consequently, T is a spanning tree of
G satisfying γpr(T ) = γpr(G).
By our earlier convention, if G is a graph of order n and γpr(G) = 2, then st−γpr(G) = n
since removing all vertices from the graph G produces the null graph with paired domi-
nation number zero. We are therefore only interested in the γ−pr-stability of graphs with
paired domination number at least 4. If G is a graph with γpr(G) ≥ 4 where x and
y are adjacent vertices in G, then D = V (G) \ {x, y} belongs to the set NI(G) and
γpr(G−D) = γpr(K2) = 2 < γpr(G). This yields the following result.
Observation 4.2. Every isolate-free graph G of order n with γpr(G) ≥ 4 satisfies
st−γpr(G) ≤ n− 2.
Proposition 4.3. If T is a spanning tree of a connected graph G such that
γpr(T ) = γpr(G), then st−γpr(T ) ≥ st
−
γpr(G).
Proof. Let S be a st−γpr -set of T . Thus, S is a set in NI(T ) with |S| = st
−
γpr(T ) such
that γpr(T − S) < γpr(T ). Since γpr(G − S) ≤ γpr(T − S) and γpr(T ) = γpr(G),
the set S is a non-isolating set of G such that γpr(G − S) < γpr(G). Hence, st−γpr(G) ≤
|S| = st−γpr(T ).
The following result shows that to determine the γ−pr-stability of a graph G, it is not
sufficient to only examine spanning trees T of G satisfying γpr(T ) = γpr(G).
A. Gorzkowska et al.: Paired domination stability in graphs 239
Proposition 4.4. For k ≥ 1 an integer, there exist connected graphs G such that
st−γpr(T )− st
−
γpr(G) = k for every spanning tree T of G with γpr(T ) = γpr(G).
Proof. For k ≥ 1, let F be obtained from two vertex disjoint copies of K2,k+1 by iden-
tifying a vertex of degree k + 1 from each copy. Let u be the resulting identified vertex
of degree 2(k + 1), and let w1 and w2 be the two vertices of degree k + 1 in F . Fur-
ther, let vi be a common neighbor (of degree 2) of u and wi for i ∈ [2]. Let G be ob-
tained from F by adding a leaf neighbor xi to wi for i ∈ [2]. Thus, diam(G) = 6 and
x1w1v1uv2w2x2 is a shortest path in G of length 6. The graph G satisfies γpr(G) = 4.
We remark that only connected graphs of diam(G) ≤ 3 have γpr(G) = 2. Therefore,
st−γpr(G) ≥ 3. Moreover, the set S = {w1, x1, x2} is a non-isolating set of minimum car-
dinality satisfying γpr(G − S) = 2 < γpr(G), and so st−γpr(G) = 3. However, the vertex
u must have degree 2 in every spanning tree T of G for which γpr(T ) = γpr(G) = 4,
implying that the vertices w1 and w2 each have k + 1 leaf neighbors in T . This implies
that every non-isolating set of T that decreases the paired domination number contains at
least k + 3 vertices. The set S = NT [w1] is a non-isolating set of minimum cardinality
satisfying γpr(T − S) = 2 < γpr(T ), and so st−γpr(T ) ≤ |S| = k + 3. Consequently,
st−γpr(T ) = k + 3, and so st
−
γpr(T )− st
−
γpr(G) = k.
Proposition 4.5. If S is a st−γpr -set of a connected isolate-free graph G with γpr(G) ≥ 4,
then γpr(G− S) = γpr(G)− 2.
Proof. Let S be a st−γpr -set of G. Suppose, to the contrary, that γpr(G− S) ≤ γpr(G)− 4.
By the connectivity of G, there exists a vertex u ∈ S that has a neighbor in the set V (G)\S.
We now consider the set S′ = S\{u}. Let D be a γpr-set of G−S. If u has a neighbor in D,
then D is a γpr-set of G−S′, implying that γpr(G−S′) ≤ |D| = γpr(G−S) ≤ γpr(G)−4,
contradicting our choice of the set S. Hence, u has no neighbor in D. Let v be an arbitrary
neighbor of u that belongs to V (G) \ S. The set D ∪ {u, v} is a PD-set of G − S′ with u
and v paired, and with the pairings of the vertices of D unchanged from their pairings in
G−S. Hence, γpr(G−S′) ≤ |D|+2 ≤ γpr(G)− 2, once again contradicting our choice
of the set S.
5 Paths and cycles
It is well known (see, for example, [11]) that for n ≥ 3 we have γpr(Cn) = γpr(Pn) =
2⌈n4 ⌉. In this section, we determine the paired domination stability of paths and cycles.
The proofs require a detailed case analysis, which is straightforward albeit tedious. We
therefore omit the proofs in this section. The γ−pr-stability of a path Pn and a cycle Cn on
n vertices is given by the following result.
Theorem 5.1. If G is a path Pn, for n ≥ 2, or a cycle Cn, for n ≥ 3, then
st−γpr(G) =

1 when n ≡ 1 (mod 4)
2 when n ≡ 2 (mod 4)
3 when n ≡ 3 (mod 4)
4 when n ≡ 0 (mod 4).
Next we determine the γ+pr-stability of a path Pn. For n ≤ 10 with n ̸= 8 and for
n = 13, no non-isolating set of vertices in a path Pn exists whose removal increases the
240 Ars Math. Contemp. 22 (2022) #P2.04 / 231–248
paired domination number, and hence, by definition, st+γpr(Pn) = ∞ for such values of n.
It is therefore only of interest to determine the γ+pr-stability of a path Pn, where n ≥ 8 and
n /∈ {9, 10, 13}.
Theorem 5.2. For n ≥ 8 and n /∈ {9, 10, 13},
st+γpr(Pn) =
{
1 when n (mod 4) ∈ {0, 3}
2 when n (mod 4) ∈ {1, 2}.
As a consequence of Theorems 5.1 and 5.2, the paired domination stability of a path is
determined.
Corollary 5.3. For n ≥ 2,
stγpr(Pn) =

1 when n (mod 4) ∈ {0, 1, 3} and n /∈ {3, 4, 7}
2 when n ≡ 2 (mod 4)
3 when n ∈ {3, 7}
4 when n = 4.
We next consider the γ+pr-stability of a cycle Cn. As shown in Theorem 5.1, the γ
−
pr-
stability of a path and a cycle of the same order are equal. This is not always the case for
the γ+pr-stability of a path and a cycle. For example, st
+
γpr(P12) = 1 and st
+
γpr(C12) = 2.
Analogously as in the case of paths, for small values of the order of a cycle the γ+pr-stability
is infinite. Namely, for n ≤ 14 with n ̸= 12 and n = 17 we have that st+γpr(Cn) = ∞. The
following result determines the γ+pr-stability of a cycle of large order.
Theorem 5.4. For n ≥ 12 and n /∈ {13, 14, 17},
st+γpr(Cn) =

2 when n ≡ 0 (mod 4)
3 when n (mod 4) ∈ {2, 3}
4 when n ≡ 1 (mod 4).
As a consequence of Theorems 5.1 and 5.4, the paired domination stability of a cycle
is determined.
Corollary 5.5. For n ≥ 3,
stγpr(Cn) =

1 when n ≡ 1 (mod 4)
2 when n (mod 4) ∈ {0, 2} and n /∈ {4, 8}
3 when n ≡ 3 (mod 4)
4 when n ∈ {4, 8}.
6 Trees
In this section, we first determine the γpr-stability of trees in the family F∆ and a new
family E∆.
Lemma 6.1. For ∆ ≥ 2, if T ∈ F∆, then st−γpr(T ) = 2∆.
A. Gorzkowska et al.: Paired domination stability in graphs 241
Proof of Lemma 6.1. Let T be an arbitrary tree in the family Fk,∆ for some k ≥ 2 and
∆ ≥ 2. We show that st−γpr(T ) = 2∆. The family Fk,2 consists of all paths P4k where
k ≥ 2. Therefore by Theorem 5.1, we have st−γpr(T ) = 4 = 2∆ for each T ∈ Fk,2, which
yields the desired result. Hence, we may assume that ∆ ≥ 3. We show, by induction on
k ≥ 2, that every tree T in the family Fk,∆ satisfies st−γpr(T ) = 2∆.
Suppose k = 2, and so T ∈ F2,∆ (where recall that ∆ ≥ 3). The tree T can therefore
be constructed from two vertex disjoint double stars T1 and T2, where Ti ∼= S(∆−1,∆−1)
for i ∈ [2], by selecting leaves w1 and w2 of T1 and T2, respectively, and adding the edge
w1w2 to T1 ∪ T2. Let xi and yi be the two vertices of Ti that are not leaves, where xiwi is
an edge. We note that y1x1w1w2x2y2 is a path in T . We note that γpr(T ) = 4 and the set
{x1, x2, y1, y2} is a γpr-set of T .
Let S be a st−γpr -set of G. Thus, S is a set in NI(G) with |S| = st
−
γpr(G) such that
γpr(T − S) = 2. Let R be a γpr-set of T − S, and so R is a minimum PD-set of
T − S (of cardinality 2). Since T [R] = P2, we note that T − S is a tree of diameter
at most 3. This implies that at most one of xi and y3−i belong to T − S for i ∈ [2]. Thus,
|S ∩ {xi, y3−i}| ≥ 1 for i ∈ [2].
Suppose that y1 ∈ S and x2 ∈ S. If x1 ∈ S, then all leaf neighbors of y1, x1 and x2
belong to S, while if y2 ∈ S, then all leaf neighbors of y1, y2 and x2 belong to S. In both
cases, |S| ≥ 3∆− 2 > 2∆.
Suppose that y1 ∈ S and x2 /∈ S. If y2 ∈ S, then all leaf neighbors of y1 and y2
belong to S, implying that |S| ≥ 2∆. If y2 /∈ S, then x1 ∈ S, implying that S contains all
leaf-neighbors of y1 and x1, and so |S| ≥ 2∆− 1. However if in this case |S| = 2∆− 1,
implying that diam(T − S) ≥ 4, a contradiction. Hence, |S| ≥ 2∆.
Suppose that y1 /∈ S and x2 ∈ S. Since T − S is a tree, y2 ∈ S and all leaf neighbors
of y2 and x2 belong to S, implying that |S| ≥ 2∆ − 1. However if in this case |S| =
2∆− 1, then S contains x2 and all leaf neighbors of y1, implying that diam(T − S) ≥ 4,
a contradiction. Hence, |S| ≥ 2∆. Therefore, in all three cases we have |S| ≥ 2∆, as
desired. This proves the base case when k = 2.
For the inductive hypothesis, let k ≥ 3 and assume that if T ′ ∈ Fk′,∆ where 2 ≤ k′ <
k, then st−γpr(T
′) = 2∆. We now consider a tree T in the family Fk,∆. Therefore, the
tree T can be constructed from k vertex disjoint double stars H1, . . . ,Hk, where Hi ∼=
S(∆− 1,∆− 1) for i ∈ [k], by selecting one leaf yi from each double star Hi and adding
k−1 edges between vertices in {y1, . . . , yk} in such a way that the resulting graph is a tree
with maximum degree ∆. Let wi and xi be the two (adjacent) vertices of Hi that are not
leaves for i ∈ [k], where yi is a leaf neighbor of xi for i ∈ [k]. We note that γpr(T ) = 2k
and the set ∪ki=1{wi, xi} is the unique γpr-set of T .
Let U be the graph of order k whose vertices correspond to the k double stars
H1, . . . ,Hk where two vertices are adjacent in U if and only if the corresponding dou-
ble stars are joined by an edge in T . We call U the underlying graph of T . By construction,
the graph U is a tree, noting that T is a tree. Let V (U) = {u1, . . . , uk} where ui is the
vertex of U corresponding to the double star Hi for i ∈ [k]. Renaming the double stars if
necessary, we may assume that u1 is a leaf in U , and that H1 is joined to H2 in T . Thus,
y1y2 ∈ E(T ) and y1yj /∈ E(T ) for j ∈ [k] \ [2]. We note that w1x1y1y2x2w2 is a path
in T . Let T ′ = T −V (H1). By construction, the tree T ′ belongs to the family Fk′,∆ where
k′ = k − 1 ≥ 2. By induction, we have st−γpr(T
′) = 2∆.
Let S be a st−γpr -set of T . Thus, S is a set in NI(T ) with |S| = st
−
γpr(T ) such that
γpr(T − S) ≤ γpr(T ) − 2 = 2k − 2. Let Q be a γpr-set of T − S, and so |Q| ≤ 2k − 2.
242 Ars Math. Contemp. 22 (2022) #P2.04 / 231–248
Let Q′ = Q ∩ V (T ′) and S′ = S ∩ V (T ′). For i ∈ [k], let Qi = Q ∩ V (Hi) and
Si = S ∩ V (Hi). We proceed further with the following claim.
Claim 6.2. |S| ≥ 2∆.
Proof of Claim 6.2. Suppose, to the contrary, that |S| ≤ 2∆− 1.
Subclaim 6.2.1. |Q1| ≥ 2.
Proof of Subclaim 6.2.1. Suppose, to the contrary, that |Q1| ≤ 1. Suppose that Q1 = ∅.
In this case, V (H1) \ {y1} ⊆ S1. If y1 ∈ S1, then |S1| = 2∆ > |S|, a contradiction.
Hence, y1 /∈ S1, and so 2∆ − 1 ≥ |S| ≥ |S1| = 2∆ − 1, implying that S = S1 and
|S| = 2∆− 1. In this case, a γpr-set of T − S contains at least one of y1 and y2. Since the
set ∪ki=2{wi, xi} is the unique γpr-set of T ′, a γpr-set of T − S is therefore not a γpr-set
of T ′, and so γpr(T − S) ≥ γpr(T ′) + 2 = 2(k − 1) + 2 = 2k, a contradiction. Hence,
|Q1| ≥ 1.
By supposition, |Q1| ≤ 1. Consequently, |Q1| = 1, implying that Q1 = {y1} and
V (H1) \ {x1, y1} ⊆ S1, and so |S1| ≥ 2∆ − 2. If x1 ∈ S1, then |S1| = 2∆ − 1
and we end up in the previous case, which leads to a contradiction. Hence, x1 /∈ S1 and
x1 /∈ Q1, implying that y2 ∈ Q with the vertices y1 and y2 paired in Q, and |S1| = 2∆−2.
By supposition, |S| ≤ 2∆ − 1. If |S| = 2∆ − 2, then S = S1 and γpr(T − S) ≥
γpr(T
′)+2 = 2k, a contradiction. Hence, |S| = 2∆−1, and so the set S contains a vertex
v′ ∈ V (T ′)\{y2}. However noting that ∆ ≥ 3, every non-isolating set of vertices of T ′−y2
that decreases the paired domination number cannot contain only one vertex, implying that
γpr(T − S) ≥ |{y1, y2}|+ γpr(T ′ − y2) = 2 + γpr(T ′) = 2k, a contradiction.
Subclaim 6.2.2. {x1, y1} ⊆ Q.
Proof of Subclaim 6.2.2. Suppose, to the contrary, that y1 /∈ Q1, implying that S′ ∈
NI(T ′). Recall that S is a st−γpr -set of T and |S
′| ≤ |S| ≤ 2∆− 1. However, st−γpr(T
′) =
2∆. Therefore, γpr(T ′−S′) ≥ γpr(T ′) = 2(k−1). Hence, γpr(T −S) = γpr(T ′−S′)+
|Q1| ≥ 2(k − 1) + 2 = 2k, a contradiction. Hence, y1 ∈ Q1.
Suppose, to the contrary, that x1 /∈ Q1. Thus, all ∆ − 2 leaf-neighbors of x1 belong
to the set S1. By Claim 6.2.1, we have |Q1| ≥ 2. Hence, the set Q1 contains w1 and one
of its leaf-neighbor w′1. We now consider the set S
∗ = S \ S1. Since S∗ ∈ NI(T ) and
(Q \ {w′1}) ∪ {x1} is a PD-set of T − S∗, we have γpr(T − S∗) ≤ |Q| = γpr(T − S),
contradicting our choice of the set S. Hence, x1 ∈ Q1.
Subclaim 6.2.3. w1 /∈ Q1.
Proof of Subclaim 6.2.3. Suppose, to the contrary, that w1 ∈ Q1. Hence, {w1, x1, y1} ⊆
Q1, and so S ∩ V (H1) = ∅ by the minimality of S. Thus, S = S′ and therefore
|S′| ≤ 2∆− 1.
We show firstly that x1 and y1 are paired in Q. Suppose, to the contrary, that x1 and
y1 are not paired in Q. This implies that y2 ∈ Q, and that y1 and y2 are paired in Q.
Suppose that x2 /∈ S, implying that S′ ∈ NI(T ′). By the minimality of the set Q, we have
x2 /∈ Q. Thus, the set Q′ ∪ {x2} is a PD-set of T ′ − S′, and so |Q′|+ 1 = |Q′ ∪ {x2}| ≥
γpr(T
′ − S′) ≥ γpr(T ′) = 2(k − 1). Hence, |Q| = |Q1| + |Q′| ≥ 3 + (2k − 3) = 2k =
γpr(T ), a contradiction. Hence, x2 ∈ S. We now consider the set S∗ = S \ {x2}. We note
that S∗ is a non-isolating set of vertices of T , and the set Q is a PD-set of T − S∗. Thus,
A. Gorzkowska et al.: Paired domination stability in graphs 243
γpr(T − S∗) ≤ |Q| ≤ 2k− 2, which contradicts our choice of the set S. Hence, x1 and y1
are paired in Q.
Since x1 and y1 are paired in Q, the vertex w1 is paired with one of its leaf neighbors,
say w′1. By the minimality of Q we note that Q1 = {w1, w′1, x1, y1}. If x2 ∈ Q, then the
set Q\{w′1, y1} is a PD-set of T −S (with w1 and x1 paired), contradicting the minimality
of Q. Hence, x2 /∈ Q. This in turn implies that y2 /∈ Q. If y2 ∈ S, then once again
we contradict the minimality of Q. Therefore, y2 /∈ S. We remark, though, that possibly
x2 ∈ S. Recall that by our earlier observations, S = S′.
Let S′′ = S\{x2}. Thus, if x2 /∈ S, then S′′ = S, while if x2 ∈ S, then S′′ = S\{x2}.
The set S′′ is a non-isolating set of T ′ such that |S′′| ≤ |S| ≤ 2∆ − 1. As observed
earlier, y2 /∈ Q′ and x2 /∈ Q′. The set Q′ ∪ {y2, x2} is a PD-set of T ′ − S′′, implying
that |Q′| + 2 ≥ γpr(T ′ − S′′) ≥ γpr(T ′) = 2(k − 1). Hence, |Q′| ≥ 2k − 4, and so
|Q| = |Q1|+ |Q′| ≥ 4 + (2k − 4) = 2k, contradicting the fact that |Q| ≤ 2k − 2.
Proof of Claim 6.2, continued: By Claim 6.2.3, w1 /∈ Q1. This implies that Q1 = {x1, y1}.
The set S1 therefore consists of the ∆− 1 leaf neighbors of w1, and so |S1| = ∆− 1. This
is true for every leaf in the tree U . Hence, if ui is a leaf in U for some i ∈ [k], then in the
corresponding double star Hi of T we have Qi = {xi, yi} and |Si| = ∆− 1. Further, the
set Si consists of the ∆− 1 leaf neighbors of wi. In particular, |Q1| = 2 and |S1| = ∆− 1.
Since the underlying tree U of T has order k ≥ 3, there are at least two leaves in U . Thus,
up is a leaf in U for some p ∈ [k] \ {1}, implying that |Qp| = 2 and |Sp| = ∆− 1.
If |Qi| ≥ 2 for all i ∈ [k], then |Q| ≥ 2k, a contradiction. Hence, |Qq| ≤ 1
for some q ∈ [k]. By our earlier observations, uq is not a leaf in the tree U , and so
q /∈ {1, p}. If |Qq| = 0, then {wq, xq} ⊆ Sq , and so |Sq| ≥ 2 (in fact, |Sq| ≥ 2∆ − 1)
and |S| ≥ |S1| + |Sp| + |Sq| ≥ (∆ − 1) + (∆ − 1) + 2 = 2∆, a contradiction. Hence,
|Qq| = 1, implying that Qq = {yq} and wq ∈ Sq , and so |Sq| ≥ 1. Since the paired dom-
inating number is an even integer and |Q| ≤ 2k, there exists r ∈ [k] \ {1, p, q} such that
|Qr| = 1. Therefore, Qr = {yr} and |Sr| ≥ 1. Hence, |S| ≥ |S1|+ |Sp|+ |Sq|+ |Sr| ≥
(∆−1)+(∆−1)+1+1 = 2∆, a contradiction. This completes the proof of Claim 6.2.
Proof of Lemma 6.1, continued: By Claim 6.2, we have |S| ≥ 2∆. By our choice of the set
S, this implies that st−γpr(T ) = |S| ≥ 2∆. Conversely, if we consider the set S = V (H1),
then S ∈ NI(T ) satisfies |S| = 2∆ and γpr(T − S) = γpr(T ′) = 2k − 2 < γpr(T ),
and so st−γpr(T ) ≤ 2∆. Consequently, st
−
γpr(T ) = 2∆. This completes the proof of
Lemma 6.1.
We determine next the γ+pr-stability of a tree in the family F∆.
Lemma 6.3. For ∆ ≥ 2, if T ∈ F∆, then st+γpr(T ) ≤ ∆− 1.
Proof. Let T be an arbitrary tree in the family Fk,∆ for some k ≥ 2 and ∆ ≥ 2. We
use the same notation as in the proof of Lemma 6.1. In particular, γpr(T ) = 2k and
H1 corresponds to a leaf u1 in the underlying tree U of T . Moreover, y1y2 is the edge
joining H1 and H2 in T . Also, wi and xi are the support vertices in the double star Hi
and wixiyi is a path in Hi for i ∈ [k]. Let L be the set of ∆ − 2 leaf neighbors of x1
in T , and let S = L ∪ {x1}. We resulting set S ∈ NI(T ) and the forest T − S has two
components, say F1 and F2 where w1 ∈ V (F1) and y1 ∈ V (F2). Moreover, γpr(T −S) =
γpr(F1) + γpr(F2) = 2 + 2k > γpr(T ). Therefore, st+γpr(T ) ≤ |S| = ∆− 1.
244 Ars Math. Contemp. 22 (2022) #P2.04 / 231–248
Recall that by Proposition 3.1, for ∆ ≥ 3, if T ∈ H∆, then stγpr(T ) = 2∆−1. Further
we remark that st+γpr(T ) = ∞. We next define another family of trees T with maximum
degree ∆ such that st−γpr(T ) = 2∆− 1. For integers ∆ ≥ 3 and ∆− 1 ≥ k ≥ 3, let Ek,∆
be a graph obtained from the path P2 with vertices u and v and the disjoint union of 2k
double stars S(∆ − 1,∆ − 1) by selecting one leaf from each double star and identifying
half of the selected leaves with the vertex v and the other half of the selected leaves with
the vertex u (see Figure 4). Let
E∆ =
⋃
k≥3
Ek,∆.
 
 







 







k
















k
Figure 4: A tree Ek,5 from the family E5.
If T is a tree from the family E∆, then st−γpr(T ) = 2∆−1. Moreover, if T is isomorphic
to the graph Ek,∆, then st+γpr(T ) = k(∆− 1). In contrast to the family H∆, the trees from
the family E∆ have finite γ+pr-stability.
7 Proof of Theorem 2.2
In this section we present a proof of Theorem 2.2, which we restate below.
Theorem 2.2. If T is a tree with maximum degree ∆ satisfying γpr(T ) ≥ 4, then the
following hold.
(a) st−γpr(T ) ≤ 2∆, with equality if and only if T ∈ F∆.
(b) stγpr(T ) ≤ 2∆− 1, and this bound is sharp for all ∆ ≥ 2.
Proof. We first prove the statement given in part (a). Since γpr(T ) ≥ 4, we have ∆ ≥ 2.
If ∆ = 2, then G is a path Pn of order n ≥ 5. In this case, the family Fk,∆ = {Pn : n ≡
0 (mod 4) and n ≥ 8}, and Theorem 5.1 and Lemma 6.1 imply the desired result. Suppose,
therefore, that ∆ ≥ 3. The sufficiency of part (a) follows from Lemma 6.1. To prove
the necessity, let T be a tree with maximum degree ∆ ≥ 3 satisfying γpr(T ) ≥ 4. Let
d = diam(T ), and so d ≥ 4. Let P : v0v1 . . . vd be a diametral path in G. Thus, v0 and vd
are leaves in T and d(v0, vd) = diam(G). We now consider the tree T rooted at the vertex
vd. Let D be a γpr-set of T .
Suppose that there is a child u1 of v2 that is a support vertex in T where u1 ̸= v1. Let
u0 be a leaf neighbor of u1. Since every PD-set of T contains all support vertices, we have
{v1, u1} ⊂ D. Renaming vertices if necessary, we may assume that u0 and u1 are paired
in D. Thus, if S consists of the vertex u1 and all leaf neighbors of u1, then S ∈ NI(T ) and
γpr(T − S) ≤ |D| − 2 = γpr(T ) − 2. Hence, st−γpr(T ) ≤ |S| ≤ ∆ < 2∆ − 1, and the
desired result follows. Assume, therefore, that every child of v2 different from v1 is a leaf.
A. Gorzkowska et al.: Paired domination stability in graphs 245
Suppose that there is a γpr-set, D2,3, of T such that v2 and v3 are paired in D2,3.
Necessarily, v1 ∈ D2,3 and v1 is paired in D2,3 with one of its leaf neighbors. Let S
consist of the vertex v1 and all of its leaf neighbors. Thus, S ∈ NI(T ) and γpr(T − S) ≤
|D2,3| − 2 = γpr(T ) − 2 < γpr(T ), implying that st−γpr(T ) ≤ |S| ≤ ∆ < 2∆ − 1, once
again implying the desired result. Therefore, we may assume that in every γpr-set of T the
vertices v2 and v3 are not paired.
Suppose that there is a γpr-set, D3, of T which contains a neighbor of v3 differ-
ent from v2. In this case, if S consists of the vertex v2 and all its descendants, then
|S| ≤ 2∆ − 1, S ∈ NI(T ) and γpr(T − S) ≤ |D3| − 2 = γpr(T ) − 2 < γpr(T ),
noting that the set D3 \ S is a PD-set of T − S and, by the minimality of D3 we have
|D3 ∩ S| = 2. Thus, st−γpr(T ) ≤ |S| ≤ 2∆− 1, and the desired result follows. Hence, we
may assume that every γpr-set of T contains the vertex v2 but no other vertex in N [v3]. In
particular, N [v3] ∩D = {v2}.
Suppose that dT (v1) < ∆ or dT (v2) < ∆. Thus, dT (v1)+ dT (v2) ≤ 2∆− 1. In order
to dominate the vertex v0, we have v1 ∈ D. By our earlier assumptions, v2 ∈ D and every
child of v2 different from v1 is a leaf. Thus by the minimality of the set D, the vertex v1 is
the only descendant of v2 that belongs to the set D, and the vertices v1 and v2 are paired in
D. Hence, if S = N [v2]∪N [v1], then S ∈ NI(T ) and |S| = dT (v1) + dT (v2) ≤ 2∆− 1.
Further, D \ {v1, v2} is a PD-set of T − S, and so γpr(T − S) ≤ |D| − 2 = γpr(T ) − 2,
implying that st−γpr(T ) ≤ |S| ≤ 2∆−1, yielding the desired result. Hence, we may assume
that dT (v1) = dT (v2) = ∆.
Suppose that dT (v3) ≥ 3, and let u2 be a child of v3 different from v2. If u2 is a leaf,
then v3 belongs to every γpr-set of T , while if u2 is not a leaf, then from the structure
of the rooted tree T the vertex u2 can be chosen to belong to some γpr-set of T . In both
cases, we contradict our earlier assumption that every γpr-set of T contains the vertex v2
but no other vertex in N [v3]. Hence, dT (v3) = 2. We now let S = N [v1] ∪ N [v2], and
so S ∈ NI(T ) and |S| = dT (v1) + dT (v2). By our earlier observations, |S| = 2∆ and
γpr(T − S) = γpr(T )− 2 < γpr(T ), implying that st−γpr(T ) ≤ |S| = 2∆. This proves the
desired upper bound.
We show next that if we have equality in the upper bound in part (a), then T ∈ F∆.
Let st−γpr(T ) = 2∆. By our earlier observations, we have that every child of v2 different
from v1 is a leaf. Further, dT (v1) = dT (v2) = ∆ and dT (v3) = 2. We now re-root the
tree T at the vertex v0, thereby interchanging the roles of v0 and vd. Identical arguments
as before show that every child of vd−2 different from vd−1 is a leaf. Further, dT (vd−1) =
dT (vd−2) = ∆ and dT (vd−3) = 2. In particular, d ≥ 6.
Suppose that d = 6, and so vd−3 = v3. In this case, the tree T is determined
and γpr(T ) = 4. Letting S = (N [v1] ∪ N [v2]) \ {v3}, we have S ∈ NI(T ) and
|S| = dT (v1) + dT (v2) − 1 = 2∆ − 1. Further, γpr(T − S) = 2 < γpr(T ). There-
fore, st−γpr(T ) ≤ |S| = 2∆− 1, a contradiction. Hence, d ≥ 7, and so vd−3 ̸= v3.
We now consider the tree T ′ = T − (N [v1] ∪ N [v2]). If γpr(T ′) = 2, then by our
earlier observations, we have d = 7 and T ′ ∼= S(∆ − 1,∆ − 1) where vd−1 and vd−2
are the two (adjacent) vertices in T ′ that are not leaves. Therefore, T ∈ T2,∆, and so T ∈
T∆. Hence, we may assume that γpr(T ′) ≥ 4, for otherwise the desired characterization
follows. In particular, d ≥ 8. As observed earlier, dT (vd−1) = dT (vd−2) = ∆, implying
that ∆(T ′) = ∆ and st−γpr(T
′) ≤ 2∆.
Let D be a γpr-set of T . Since every PD-set of T contains the set of support vertices,
we note that v1, v2 ∈ D. By the minimality of D, no leaf-neighbor of v1 or v2 belongs to
246 Ars Math. Contemp. 22 (2022) #P2.04 / 231–248
D. If v3 ∈ D, then v4 ∈ D (with v3 and v4 paired in D). However in this case, we can
replace v3 in D with an arbitrary neighbor of v4 that does not belong to D. Hence, we can
choose the γpr-set D of T so that v3 /∈ D. The resulting set D when restricted to V (T ′) is
a PD-set of T ′, implying that γpr(T ′) ≤ |D| − 2 = γpr(T )− 2. Conversely, every PD-set
of T ′ can be extended to a PD-set of T by adding to it the vertices v1 and v2 (with v1 and
v2 paired), and so γpr(T ) ≤ γpr(T ′) + 2. Consequently, γpr(T ) = γpr(T ′) + 2.
Suppose that st−γpr(T
′) < 2∆. Let S′ be a st−γpr -set of T
′. Thus, S is a set in NI(T ′)
with |S′| = st−γpr(T
′) < 2∆ such that γpr(T −S′) < γpr(T ′). If D′ is a γpr-set of T ′−S′,
then D′∪{v1, v2} is a PD-set of T−S, and so γpr(T−S′) ≤ |D′|+2 = γpr(T−S′)+2 <
γpr(T
′) + 2 = γpr(T ). Hence, S′ ∈ NI(T ) and γpr(T − S′) < γpr(T ′), implying that
st−γpr(T ) ≤ |S
′| = st−γpr(T
′) < 2∆, a contradiction. Therefore, st−γpr(T
′) = 2∆.
Hence, the tree T ′ satisfies ∆(T ′) = ∆, γpr(T ′) ≥ 4 and st−γpr(T
′) = 2∆. Proceeding
by induction, we have T ′ ∈ F∆. Thus, T ′ is constructed from the disjoint union of k′
double stars each isomorphic to S(∆− 1,∆− 1), by selecting one leaf from each double
star and adding k′−1 edges between these selected leaves to produce a tree with maximum
degree ∆. The resulting tree T ′ satisfies γpr(T ′) = 2k′ with the 2k′ support vertices
forming a γpr-set of T ′.
By construction of T ′, the tree T ′ contains the vertex v4 but not the vertex v3. Sup-
pose that v4 is a support vertex in T ′, implying by construction of T ′ that v4 is a vertex
of degree ∆ in T ′. Let S = (N [v1] ∪ N [v2]) \ {v3}. We note that S ∈ NI(T ) and
|S| = 2∆ − 1. Let D′ be the (unique) γpr-set of T ′, and so D′ is the set of 2k′ support
vertices in T ′. In particular, we note that v4 ∈ D′. The set D′ is a PD-set of T − S, and
so γpr(T − S) ≤ |D′| = γpr(T ′) = γpr(T ) − 2. Therefore, st−γpr(T ) ≤ |S| = 2∆ − 1,
a contradiction. Hence, v4 is a leaf of T ′, and so v4 is a leaf in one of the k′ double stars in
the construction of T ′. Selecting the leaf v4 from this double star and selecting the leaf v3
from the double star induced by N [v1] ∪N [v2], which is isomorphic to S(∆− 1,∆− 1),
and adding back the edge v3v4 we re-construct the tree T , showing that T ∈ F∆. This
completes the proof of part (a).
Part (b) now follows readily from part (a). If T ∈ F∆ for some ∆ ≥ 2, then by Lem-
mas 6.1 and 6.3, we have stγpr(T ) ≤ ∆− 1. Hence, we may assume that T /∈ F∆ for any
∆ ≥ 2, for otherwise the bound in part (b) is immediate. With this assumption, the upper
bound in part (b) follows immediately from part (a) noting that stγpr(T ) ≤ st−γpr(T ) ≤
2∆− 1. That the bound is tight for all ∆ ≥ 2 follows from Proposition 3.1.
8 Proof of Theorem 2.3
In this section we present a proof of Theorem 2.3, which we restate below.
Theorem 2.3. If G is a connected graph with γpr(G) ≥ 4, then st−γpr(G) ≤ 2∆(G), and
this bound is sharp.
Proof. Let G be a connected graph with γpr(G) ≥ 4 and let ∆ = ∆(G). Since
γpr(G) ≥ 4, we have ∆ ≥ 2. If ∆ = 2, then G is a path Pn or a cycle Cn, and
by Theorem 5.1, we have st−γpr(G) ≤ 2∆, with equality if and only if n ≡ 0 (mod 4).
Assume, therefore, that ∆ ≥ 3.
Let T be a spanning tree of G such that γpr(T ) = γpr(G). We note that such a tree ex-
ists by Lemma 4.1. Let S be a st−γpr -set of T . Thus, S is a set in NI(T ) with
|S| = st−γpr(T ) such that γpr(T − S) < γpr(T ). By Observation 4.2, we have
A. Gorzkowska et al.: Paired domination stability in graphs 247
|S| = st−γpr(T ) ≤ n − 2. Since S ∈ NI(T ), every vertex in T − S, and therefore in
the supergraph G − S, has degree at least 1. Hence, S ∈ NI(G) and since γpr(G − S) ≤
γpr(T − S), we have γpr(G − S) < γpr(G). Thus, st−γpr(G) ≤ |S| = st
−
γpr(T ). By The-
orem 2.2, we have st−γpr(T ) ≤ 2∆(T ). Noting that ∆(T ) ≤ ∆(G), we therefore have that
st−γpr(G) ≤ st
−
γpr(T ) ≤ 2∆(T ) ≤ 2∆(G) = 2∆.
To show that the upper bound in Theorem 2.3 is tight, we present a family of graphs
with maximum degree ∆ and γpr(G) ≥ 4 satisfying st−γpr(G) = 2∆. Our first family, G∆,
is constructed as follows. For k ≥ 2 and ∆ ≥ 2, let Gk,∆ be a graph obtained from k
double stars S(∆ − 1,∆ − 1) by choosing two leaves at distance 3 apart in each double
star and adding k edges between the chosen leaves in such a way, that every chosen vertex
has degree 2 in the resulting graph. Let G∆ be the family of all such graphs Gk,∆ for
all k ≥ 2. The graph G2,6 ∈ G6, for example, is illustrated in Figure 5. We note that
γpr(Gk,∆) = 2k and that set of 2k vertices of degree ∆ is the unique γpr-set of Gk,∆.
Furthermore, st−γpr(Gk,∆) = 2∆.



 



















Figure 5: The graph G2,6 from a class of graphs Gk,∆.
Recall that by definition we have stγpr(G) ≤ st−γpr(G) for every graph G. Hence, as an
immediate consequence of Theorem 2.3 we have Corollary 2.4. Recall its statement.
Corrolary 2.4. If G is a connected graph with γpr(G) ≥ 4, then stγpr(G) ≤ 2∆(G).
It remains an open problem, however, to determine if the upper bound of Corollary 2.4
is best achievable for all values of possible value of ∆(G) = ∆ ≥ 2. If ∆ = 2 and G is
a path, then G ∼= Pn where n ≥ 5, and stγpr(G) ≤ 2∆ − 2 by Corollary 5.3. If ∆ = 2
and G is a cycle, then G ∼= Cn where n ≥ 5, and stγpr(G) ≤ 2∆ by Corollary 5.5,
with equality if and only if G = C8. Hence, the only connected graph G with maximum
degree ∆ = 2 satisfying γpr(G) ≥ 4 and stγpr(G) = 2∆ is the 8-cycle, namely G = C8.
For ∆ ≥ 3, we do not know of a connected graph G with maximum degree ∆ satisfying
γpr(G) ≥ 4 and stγpr(G) = 2∆.
By Corollary 5.5 and Proposition 3.1, for any given ∆ ≥ 2, there do exists infinite
families of connected graphs G with maximum degree ∆ satisfying stγpr(G) = 2∆ − 1.
Thus, if the upper bound of Corollary 2.4 can be improved to stγpr(G) ≤ 2∆ − 1 in the
case when ∆ ≥ 3, then this bound would be tight.
ORCID iDs
Aleksandra Gorzkowska https://orcid.org/0000-0001-5335-7351
Michael A. Henning https://orcid.org/0000-0001-8185-067X
248 Ars Math. Contemp. 22 (2022) #P2.04 / 231–248
Monika Pilśniak https://orcid.org/0000-0002-3734-7230
Elżbieta Tumidajewicz https://orcid.org/0000-0002-1413-2413
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.05 / 249–269
https://doi.org/10.26493/1855-3974.1955.1cd
(Also available at http://amc-journal.eu)
Notes on weak-odd edge colorings of digraphs*
César Hernández-Cruz
Facultad de Ciencias, Universidad Nacional Autónoma de México,
Av. Universidad 3000, Circuito Exterior S/N, Ciudad Universitaria, CDMX, México
Mirko Petruševski †
Faculty of Mechanical Engineering - Skopje, University Ss Cyril and Methodius,
1000 Skopje, Macedonia
Riste Škrekovski
FMF, University of Ljubljana, 1000 Ljubljana, Slovenia, and
Faculty of Information Studies, 8000 Novo mesto, Slovenia, and
University of Primorska, FAMNIT, 6000 Koper, Slovenia
Received 18 March 2019, accepted 29 July 2021, published online 27 May 2022
Abstract
A weak-odd edge coloring of a general digraph D is a (not necessarily proper) coloring
of its edges such that for each vertex v ∈ V (D) at least one color c satisfies the following
conditions: if d−D(v) > 0 then c appears an odd number of times on the incoming edges at
v; and if d+D(v) > 0 then c appears an odd number of times on the outgoing edges at v. The
minimum number of colors sufficient for a weak-odd edge coloring of D is the weak-odd
chromatic index, denoted χ′wo(D). It is known that χ′wo(D) ≤ 3 for every digraph D, and
that the bound is sharp. In this article we show that the weak-odd chromatic index can
be determined in polynomial time. Restricting to edge colorings of D with at most two
colors, the minimum number of vertices v ∈ V (D) for which no color c satisfies the above
conditions is the defect of D, denoted def(D). Surprisingly, it turns out that the problem of
determining the defect of digraphs is (polynomially) equivalent to the problem of finding
the matching number of simple graphs. Moreover, we characterize the classes of associated
digraphs and tournaments in terms of the weak-odd chromatic index and the defect.
Keywords: Digraph, weak-odd edge coloring, weak-odd chromatic index, defective coloring, tourna-
ment.
Math. Subj. Class. (2020): 05C15, 05C20
*This work is partially supported by ARRS Program P1-0383 and ARRS Project J1-1692.
†Corresponding author.
E-mail addresses: japo@ciencias.unam.mx (César Hernández-Cruz), mirko.petrushevski@gmail.com
(Mirko Petruševski), skrekovski@gmail.com (Riste Škrekovski)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
250 Ars Math. Contemp. 22 (2022) #P2.05 / 249–269
1 Introduction
Throughout the article we mainly follow terminology and notation used in [1]. All graphs
and digraphs are considered to be finite. Loops, parallel edges and parallel arcs are admis-
sible, i.e., strictly speaking, we consider the general setup of pseudographs and directed
pseudographs, but in order to avoid lengthy terminology, we abbreviate these terms to
‘graphs’ and ‘digraphs’, respectively.
Cheilaris et al. [4] introduced the notion of odd coloring of a hypergraphH as a vertex
coloring such that for every (hyper-)edge e ∈ E(H) there is a color c with an odd number
of vertices of e colored by c. Restricting to graphs G (and thus to plain edges), the previous
coloring notion is merely the usual notion of ‘proper’ coloring of G. Through interchanging
the roles played by vertices and edges, initially motivated by [5, 6], the analogous edge
coloring notion for graphs was introduced in [10] as follows. An edge coloring of G is said
to be weak-odd if it holds that:
(WO) For every vertex v ∈ V (G) with degree dG(v) > 0, at least one color c appears an
odd number of times on the set of edges incident with v.
The additional adjective ‘weak’ has been included in the name simply in order to dif-
ferentiate this from the related, already existing, notion of ‘odd edge coloring’ of graphs,
defined in [12]. (The latter notion has stronger requirements for the colors appearing at a
vertex.)
Let us clarify that, by definition, any loop at v colored by c contributes 2 to the count of
appearances of c on EG(v) (the set of all edges incident with v). An obvious necessary and
sufficient condition for weak-odd edge colorability of graph G is the absence of ‘isolated
loops’, i.e., nonempty trivial components. A weak-odd edge coloring of G using at most
k colors is referred to as a weak-odd k-edge coloring of G, and such a graph is said to be
weak-odd k-edge colorable. The weak-odd chromatic index, χ′wo(G), is the minimum k for
which G is weak-odd k-edge colorable. Obviously, apart from ‘isolated loops’, any other
loop addition or removal does not influence the existence nor alters the value of χ′wo(G).
The following characterization of graphs G in terms of χ′wo(G) was obtained in [10]. It
makes use of the next two notions. A graph G is even (resp. odd) if every vertex v ∈ V (G)
has even (resp. odd) degree dG(x).
Theorem 1.1. For any connected graph G whose edge set does not consist entirely of
loops, it holds that
χ′wo(G) =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
0 if G =K1,
1 if G is odd,
2 if G has even order or is not even,
3 if G is even, has odd order, and is not K1.
This paper treats the analogous coloring notion for digraphs. In the next section we fur-
ther explain the motivation behind the definition of ‘weak-odd edge coloring’ of digraphs,
introduced in [11], and then show that the corresponding coloring index χ′wo can be deter-
mined in linear time. We also address a related problem concerning the minimum number
of vertices at which an arbitrary 2-edge coloring of a digraph fails at being ‘weak-odd’,
and prove a connection with the problem of determining the matching number of a simple
graph. In Section 3, the discussion restricts to two common classes of digraphs, namely,
C. Hernández-Cruz et al.: Notes on weak-odd edge colorings of digraphs 251
the class of associated digraphs1 and the class of tournaments. For each of these classes we
give a descriptive characterization of their members in terms of χ′wo. In the final section
we briefly convey our thoughts on possible further work on the topic of weak-odd edge
colorings of digraphs. For the end of this introductory section, we mention some common
notions and facts that will be frequently used throughout.
1.1 General terminology and notation
We denote the symmetric difference of sets A,B by A ⊕ B. The same notation is in use
for the symmetric difference of two spanning subgraphs A,B of a ground graph. Given a
graph G and an even-sized subset T of V (G), a spanning subgraph H is a T -join of G if
dH(v), the degree of v with respect to H , is odd for all v ∈ T and even for all v ∈ V (G)∖T .
The symmetric difference of a T ′-join and a T ′′-join of G is a T ′ ⊕ T ′′-join, which yields
the following classical result (see [14]): every connected graph G contains a T -join. In
particular, every even-ordered connected graph G has an odd factor (i.e., a V (G)-join).
An edge coloring of a graph G (resp. digraph D) with color set S is an assignment
E(G) → S (resp. A(D) → S). Every T -join of G can be interpreted as an edge coloring
with color set {1,2} such that 1 is used an odd number of times at each v ∈ T and and even
number of times (possibly zero) at each v ∈ V (G) ∖ T .
Given a digraph D and a vertex v ∈ V (D), the size of the set ∂−D(v), of incoming
edges at v, (resp. ∂+D(v), of outgoing edges at v,) is the in-degree d−D(v) (resp. out-degree
d+D(v)) of the vertex v; we call each of ∂−D(v) and ∂+D(v) (resp. d−D(v) and d+D(v)) a semi-
cut (resp. semi-degree) of v. Since loops are allowed, let us clarify that ∂−D(v) ∩ ∂+D(v)
constitutes the set of loops at v; in other words, any loop at vertex v contributes 1 to each
semi-degree of v, i.e., strictly speaking d−D(v) and d−D(v) are the semi-pseudodegrees of
v (the in-pseudodegree and out-pseudodegree, respectively). The sum dD(v) = d−D(v) +
d+D(v) is the degree of v; a vertex of degree 0 is said to be isolated. Given a nonisolated
vertex v, if d−D(v) = 0 (resp. d+D(v) = 0), then v is a source (resp. sink) of D. Any source
or sink is a peripheral vertex of D, whereas a nonisolated vertex that is neither a source
nor a sink is an intermediate vertex. A vertex u is said to dominate a vertex v if v ∈ ∂+D(u);
equivalently, v is dominated by u.
Two graphs or digraphs are considered identical if they are isomorphic to each other.
The numbers of vertices and edges of graph or digraph D are denoted by n(D) and m(D);
these basic parameters are the order and size of D, respectively; a graph or digraph of order
1 (resp. size 0) is trivial (resp. empty).
A graph is bipartite if its vertex set can be partitioned into two subsets X and Y so
that every edge has one end in X and one end in Y ; such a partition (X,Y ) is called a
bipartition of the graph, and X and Y its partite sets. We denote a bipartite graph G with
bipartition (X,Y ) by G[X,Y ]. Given a digraph D, its split (or bipartite representation),
BG(D), is the bipartite graph G whose partite sets V +, V − are copies of V (D). For each
v ∈ V (D), there is one vertex v+ ∈ V + and one v− ∈ V −. For each directed uv-edge in
D, there is an edge with endpoints u+ and v− in G. Hence, the degrees of the vertices
v+, v− in the split of D are precisely the out-degree and in-degree of v in D, respectively;
the pair (v+, v−) is obtained by splitting v in regard to D. The re-identification of each
such pair (v+, v−) into v results in the so-called underlying graph of D, denoted G(D).
1A digraph D is associated if for every arc (u, v) of D, the arc (v, u) is also present in D, and the number
of loops at any vertex is even.
252 Ars Math. Contemp. 22 (2022) #P2.05 / 249–269
Furthermore, any balanced bipartite graph G is a split of some digraph D, i.e., can be
‘transformed’ into D by reversing the described procedure.
The other way around, any graph G can be regarded as a digraph D(G), by replacing
each of its edges by two oppositely oriented arcs with the same ends (each loop of G gives
rise to two directed loops on the same vertex); this digraph is the associated digraph of G.
One may also obtain a digraph D from a graph G by replacing each edge by one arc on the
same endpoints; such a digraph D is an orientation of G. A tournament is an orientation
of a complete graph. Conversely, the underlying graph G(D) of a digraph D is obtained
by ‘forgetting orientation’. A directed path or directed cycle is an orientation of a path or
cycle in which each vertex dominates its successor in the sequence.
A digraph D is said to be strongly connected (or simply strong) if for any pair u, v
of its vertices there is a directed uv-path, i.e., a directed path joining vertex u with vertex
v. Given a digraph D, every maximal strong subdigraph of D is a strong component of
D. The condensation C(D) of a digraph D is the loopless directed (multi)graph whose
vertices correspond to the strong components of D, with any two vertices of C(D) being
linked by as many directed edges as there are directed edges in D linking the corresponding
strong components, and with the consistent orientation. The peripheral strong components
of D which correspond to the vertices of C(D) that are sources (resp. sinks), are the ini-
tial (resp. terminal) strong components; the remaining strong components of D are called
intermediate or isolated according to the nature of the corresponding vertices in C(D).
2 Weak-odd edge colorings of digraphs
In the realm of digraphs D, when defining the notion of ‘weak-odd edge coloring’ two
options come to mind. Initially, one may obtain a ‘directed version’ of the condition (WO)
as follows:
(
Ð→
W
Ð→
O) For every vertex v ∈ V (D), on each nonempty semi-cut of v at least one color c
appears an odd number of times.
However, the above ‘definition’ fails to capture the essence of digraphs since it basically
ignores the fact that arc sets ∂−D(v) and ∂+D(v) are incident with a common vertex (namely,
v). Actually, a moment’s thought reveals that, if we decide to adopt the initial definition,
then this coloring notion for digraphs would be merely a ‘disguise’ of weak-odd edge
coloring of bipartite graphs with equally sized partite sets. Namely, it can readily be seen
that an edge coloring of D satisfies condition (
Ð→
W
Ð→
O) if and only if the induced edge coloring
on BG(D) satisfies condition (WO). Consequently, in view of Theorem 1.1, every digraph
D admits a 3-edge coloring obeying (
Ð→
W
Ð→
O); moreover, three colors are indeed required if
and only if at least one component of BG(D) is a nonempty even graph of odd order. One
such example is depicted in Figure 1. Notice that if at least one of the loops is removed
from D then the obtained digraph would admit a 2-edge coloring that satisfies condition
(
Ð→
W
Ð→
O).
As noticed, unlike for graphs, in the realm of digraphs the presence of loops may in-
fluence the value of the corresponding chromatic index in a nontrivial manner. This is so
because any such loop in the digraph is no longer a loop in its split.
A more appropriate definition of the notion of ‘weak-odd edge coloring’ for digraphs,
the one we shall adopt in this study, is obtained as follows. Recall that any graph G can
be seen as a digraph, the associated digraph D(G). In the obvious fashion, every edge
C. Hernández-Cruz et al.: Notes on weak-odd edge colorings of digraphs 253
u
v
w w+ w−
v+ v−
u+ u−
Figure 1: (left) A digraph D that fails condition (
Ð→
W
Ð→
O) under any 2-edge coloring, and
(right) its split BG(D) with the left-hand (resp. right-hand) partite set representing V +
(resp. V −). The nonempty component of BG(D) is an even graph of odd order.
u u
v v
w w
Figure 2: Digraphs D1 (left) and D2 (right) are obtained from the digraph in Figure 1 by
removing a certain loop (at v and at u, respectively). Both admit 2-edge colorings fulfilling
condition (
Ð→
W
Ð→
O), but none admits a 2-edge coloring satisfying (
ÐÐ→
WO).
coloring φ of G can be interpreted as an edge coloring φD of D(G). Notice that if φ is
weak-odd then φD satisfies the condition (
ÐÐ→
WO) below, which states a stronger requirement
than the one imposed by (
Ð→
W
Ð→
O). This particular reasoning served as the motivation in [11]
for defining an edge coloring of a digraph D to be weak-odd if:
(
ÐÐ→
WO) For every vertex v ∈ V (D), at least one color c appears an odd number of times
on each nonempty semi-cut of v.
Same as with (
Ð→
W
Ð→
O), in case v is a sink (resp. source), the condition (
ÐÐ→
WO) amounts to
the appearance of c an odd number of times on the incut ∂−D(v) (resp. outcut ∂+D(v)). The
difference between (
Ð→
W
Ð→
O) and (
ÐÐ→
WO) is reflected at the intermediate vertices (cf. Figure 2).
The minimum number of colors sufficient for a weak-odd edge coloring of a digraph D
is the weak-odd chromatic index, denoted χ′wo(D). A weak-odd edge coloring of D using
at most k colors is referred to as a weak-odd k-edge coloring, and any such D is said to be
weak-odd k-edge colorable. Hence, χ′wo(D) is the minimum k for which D is weak-odd
k-edge colorable.
254 Ars Math. Contemp. 22 (2022) #P2.05 / 249–269
Interestingly, analogous to graphs, the same upper bound (of three colors) holds for the
weak-odd chromatic index of digraphs. Namely, the following was proven in [11].
Theorem 2.1. Every digraph is weak-odd 3-edge colorable.
As already illustrated through Figures 1 and 2, the bound χ′wo(D) ≤ 3 is sharp, i.e.,
not every digraph is weak-odd 2-edge colorable. Analogous to the setting of graphs, it is
quite trivial to characterize which digraphs are weak-odd 1-edge colorable. Indeed, the
inequality χ′wo(D) ≤ 1 holds if and only if for every vertex v ∈ V (D) both semi-degrees
d−D(v), d+D(v) are odd or zero. Furthermore, χ′wo(D) = 0 holds precisely when D is empty.
Thus, in order to characterize all digraphs in terms of their weak-odd chromatic index, it
suffices to figure out which are the weak-odd 2-edge colorable ones.
2.1 Characterization of weak-odd 2-edge colorability
The partial split, PS(D), of given digraph D is a graph obtained by splitting (in regard
to D) only those vertices v ∈ V (D) for which at least one semi-degree is even (including
zero), and then forgetting orientation. In other words, PS(D) is the graph obtained from
BG(D) by re-identifying each pair (u+, u−) for which both d+D(u) and d−D(u) are odd
(cf. Figure 3). In particular, if no vertex of D has only odd-sized semi-cuts, then PS(D)
is the same graph as the split BG(D); contrarily, if every nonisolated vertex of D has only
odd-sized semi-cuts, then PS(D) is the same as the underlying graph G(D). However, in
general, these three graphs related to D differ from each other.
D1 : D2 :
u+ u−
v+ v−
w+ w− w+ w
−
v
u+ u− u+ u− u
v+ v
−
w+ w−
v+ v
−
w+ w
−
Figure 3: The split BG(D1) and partial split PS(D1) (left), and the split BG(D2) and
partial split PS(D2) (right), where D1,D2 are the digraphs from Figure 2. The induced
3-partition {V1, V2, V3} of V (PS(D1)) consists of V1 = {v}, V2 = {u+, u−,w+,w−}
and V3 = ∅, whereas the corresponding 3-partition of V (PS(D2)) has V1 = {u},
V2 = {v+, v−,w+,w−}, V3 = ∅.
We distinguish between three types of vertices in PS(D) inducing a 3-partition
{V1, V2, V3} of V (PS(D)):
• the first type of vertices, constituting V1, are the members of V (D) ∩ V (PS(D)),
i.e., the vertices u of D having odd semi-degrees d+D(u), d−D(u).
• the second type of vertices, forming V2, are the members v of V (PS(D))/V (D)
that have even degree dPS(D)(v).
• finally, the third type of vertices, comprising V3, are the members w of V (PS(D))/
V (D) that have odd degree dPS(D)(w).
C. Hernández-Cruz et al.: Notes on weak-odd edge colorings of digraphs 255
For simplicity of presentation, on several occasions we shall employ the following ad-
hoc terminology and notation. Given a graph G, a vertex x ∈ V (G) is said to be even
(resp. odd) if dG(x) is even (resp. odd). The set of even (resp. odd) vertices of G is denoted
EvenV (G) (resp. OddV (G)). Note that, by the handshake lemma, OddV (G) is always
even-sized. Thus, V3 above equals OddV (PS(D)), whereas V1 ⊍ V2 = EvenV (PS(D)).
Theorem 2.2. A digraph D is weak-odd 2-edge colorable if and only if for every nonempty
component K of PS(D) we have that V (K) ∩ V2 is even-sized or V (K) ∩ V3 ≠ ∅.
Proof. Assuming χ′wo(D) ≤ 2, let φ ∶ A(D) → {1,2} be a weak-odd edge coloring of D
and consider the induced edge coloring of PS(D). Observe that for every vertex u ∈ V1,
each of the colors 1 and 2 is used an even number of times on EPS(D)(u). Indeed, the edge
set EPS(D)(u) corresponds to the entire cut ∂D(u) = ∂+D(u) ⊍ ∂−D(u); thus, since both
constituents in this disjoint union are odd-sized, for every color c ∈ {1,2} the parities of
∣∂+D(u)∩φ−1(c)∣ and ∣∂−D(u)∩φ−1(c)∣ are equal. In contrast, for every nonisolated vertex
v ∈ V2, each of the colors 1 and 2 appears an odd number of times on EPS(D)(v). The
reason behind this is that the edge set EPS(D)(v) corresponds to a nonempty even-sized
semi-cut of a vertex in D.
Therefore, if K is a nonempty component of PS(D) such that V (K) ⊆ V1 ⊍ V2,
then each of the two color classes induces in K a subgraph Ki (i ∈ {1,2}) such that
OddV (Ki) = V (K) ∩ V2. Consequently, the intersection V (K) ∩ V2 is even-sized.
Arguing in the opposite direction, assume now that every nonempty component K of
PS(D) meets the stated requirements. We construct an assignment E(K) → {1,2} as
follows. If V (K) ∩ V2 is even-sized, then define T = V (K) ∩ V2. Otherwise, select an
odd-sized subset SK ⊆ V (K) ∩ V3 and define T = (V (K) ∩ V2) ⊍ SK . Either way, T
is an even-sized subset of V (K). Therefore, there exists a T -join H of K. Color E(H)
with 1 and E(K)/E(H) with 2. Consider the induced edge coloring of D, and observe
the following.
(1) At each vertex u ∈ V (D) ∩ V1, precisely one of the colors 1,2 satisfies condition
(
ÐÐ→
WO). Indeed, by construction, each color is used an even number of times on
∂D(u), and thus has equal parities of appearance on the odd-sized sets ∂−D(u) and
∂+D(u), respectively.
(2) At each nonisolated vertex v ∈ V (D)/V1 such that the vertices v+, v− ∈ V2, colors 1
and 2 both satisfy condition (
ÐÐ→
WO). Namely, by construction, color 1 is used an odd
number of times on each nonempty semi-cut of v; on the other hand, both ∂−D(v) and
∂+D(v) are even-sized.
(3) At each vertex w ∈ V (D)/V1 such that one of the vertices w+,w− belongs in V2 (and
the other in V3), precisely one of the colors 1,2 satisfies condition (
ÐÐ→
WO). Indeed,
if the ‘half’ of w belonging to V3 is used in some SK then color 1 meets (
ÐÐ→
WO);
otherwise, color 2 does so.
Thus, the digraph D is weak-odd 2-edge colorable.
For example, the digraph D depicted in Figure 4 is weak-odd 2-edge colorable because
the only nonempty component of PS(D) intersects V3. With the notation employed in the
256 Ars Math. Contemp. 22 (2022) #P2.05 / 249–269
proof, if we take E(H) to consist of x+y−, y+u, and a uz− edge, then the induced weak-
odd 2-edge coloring of D assigns color 1 to xy, yu and a uz arc, and assigns color 2 to the
rest of A(D).
x
yz
u
x+ x−
y+ y−
z+ z−
u
Figure 4: A digraph D (left) and its partial split PS(D) (right). The induced 3-partition
has V1 = {u}, V2 = {x+, x−, y−, z−} and V3 = {y+, z+}.
Contrarily, the digraph D depicted in Figure 5 is not weak-odd 2-edge colorable, since
the induced 3-partition has V1 = {u}, V2 = {x+, x−, y+, y−, z+, z−} and V3 = ∅, the only
nonempty component of its partial split does not intersect V3 and contains an odd number
of vertices from V2.
x
yz
u
x+ x−
y+ y−
z+ z−
u
Figure 5: A digraph D (left) and its partial split PS(D) (right).
The proof of Theorem 2.2 and the fact that the problem of constructing a T -join of
any connected graph G for a given even-sized subset T of V (G) is solvable in linear time
(see [14]), imply that the decision problem of whether χ′wo(D) ≤ 2 is solvable in linear
time; in the affirmative case, a weak-odd 2-edge coloring of D can be found in linear time.
Thus, in view of Theorem 2.1 and the subsequent discussion, we conclude the following.
Corollary 2.3. The weak-odd chromatic index of any digraph D and a corresponding
weak-odd χ′wo(D)-edge coloring can be determined in linear time.
To end this subsection we point out an infinite family of digraphs having weak-odd
chromatic index equal to 3. A digraph is said to be even if every vertex v ∈ V (D) is of
C. Hernández-Cruz et al.: Notes on weak-odd edge colorings of digraphs 257
even degree dD(v); in other words, the requirement is that the semi-degrees of v are of
equal parity.
Figure 6: Two even digraphs with weak-odd chromatic index 3.
Proposition 2.4. If an even digraph D has an odd number of peripheral vertices, then
χ′wo(D) = 3.
Proof. We may assume that D is connected. Consider the partial split PS(D) of D and the
induced 3-partition {V1, V2, V3}. Obviously, V3 = ∅ and the number of isolated vertices
of PS(D) equals the number of peripheral vertices of D. However, this implies that the
number of nonisolated vertices of PS(D) which belong in V2 is odd. Therefore, an odd
number of nonempty components of PS(D) fail to meet the requirements of Theorem 2.2.
Notice that out of the two even digraphs depicted in Figure 6, only the left one has an
odd number of peripheral vertices; nevertheless, neither is weak-odd 2-edge colorable. This
demonstrates that the condition ‘an odd number of peripheral vertices’ from the statement
of Proposition 2.4, although sufficient, is by no means necessary.
2.2 Defective weak-odd 2-edge colorings
The following straightforward result serves as our motivation for the brief discussion within
this subsection. In a way, it tells that every graph can be almost weak-odd 2-edge colored.
Proposition 2.5. Every connected graph G admits a 2-edge coloring such that condition
(WO) is satisfied at each vertex apart from a prescribed vertex v ∈ V (G).
Proof. We construct an even-sized subset T of V (G) as follows. If n(G) is even, then
define T = V (G); otherwise, let T = V (G)/{v}. Since G is connected, consider a T -join
H of G. Color E(H) with 1, and the rest of E(G) with 2. The obtained 2-edge coloring
of G clearly fulfills condition (WO) at each vertex differing from v.
One naturally wonders if there exists an analogous result for digraphs that bounds (pre-
sumably with 1) the number of ‘defective vertices’, in regard to condition (
ÐÐ→
WO), for some
2-edge coloring? Unfortunately, in contrast to graphs, there are connected digraphs such
that for any 2-edge coloring the condition (
ÐÐ→
WO) fails at an unbounded number of vertices.
To construct examples, consider as a ‘gadget digraph’ D the right-hand digraph from
Figure 6. Observe that no 2-edge coloring of D can fulfill condition (
ÐÐ→
WO) at all vertices
258 Ars Math. Contemp. 22 (2022) #P2.05 / 249–269
excepting the sink (or the source). Thus, by taking any number, say n, of copies of D
and identifying their sinks (cf. Figure 7) we obtain a connected digraph D′n such that under
any 2-edge coloring at least n of its vertices are ‘defective’ in regard to condition (
ÐÐ→
WO).
A similar construction using the same gadget graph shows that even strong connectedness
comes to no avail in this regard. Namely, take an arbitrary number n ≥ 2 of copies of D,
arrange them in circular order and then identify pairwise the corresponding sink and source
of each neighboring copies (cf. Figure 7). Once again, the obtained strong digraph D′′n
under any 2-edge coloring has at least n ‘defective’ vertices in regard to condition (
ÐÐ→
WO).
The following question comes to mind: Does anything change in regard to this problem if
we confine to simple digraphs, or even more so, to digraphs with simple underlying graphs?
The next result answers the question in negative.
Figure 7: The digraph D′3 (left), and the digraph D
′′
6 (right).
Proposition 2.6. For any given positive integer n, there exists a strongly connected digraph
D with simple underlying graph G(D) such that under any 2-edge coloring of D at least
n of its vertices are ‘defective’ in regard to condition (
ÐÐ→
WO).
Proof. Simply take D to be a complete subdivision of D′′n. In other words, subdivide (at
least once) each arc e ∈ A(D′′n) and orient the newly formed arcs consistently with e.
Given a digraph D, let def(D), the defect of D, denote the minimum number of ‘defec-
tive vertices’ in regard to condition (
ÐÐ→
WO) taken over all 2-edge colorings of D. A question
that naturally arises is whether this parameter can be effectively determined. As it turns
out, the parameter def(D) is closely related to yet another graph construction, relating a
simple graph GD to each digraph D, which we describe next.
Start from the induced subgraph BC(D) ⊆ PS(D) that consists of the ‘bad compo-
nents’ of PS(D) in regard to the requirement of Theorem 2.2; in other words,
V (BC(D)) = ⋃K V (K), where the union is taken over all components K of PS(D)
such that V (K) ∩ V2 is odd-sized and V (K) ∩ V3 is empty.
Thus, the vertex set of GD consists of vertices vK corresponding to components K of
BC(D). As for the edge set of GD, make two distinct vertices vK′ and vK′′ adjacent if
the respective bad components K ′ and K ′′ contain the ‘halves’ v+ and v− of some vertex
C. Hernández-Cruz et al.: Notes on weak-odd edge colorings of digraphs 259
v ∈ V (D). To exemplify, we shall make use of the digraphs D′n and D′′n defined above. For
the first of these digraphs, it is readily seen that each of the n+ 1 nonempty components of
PS(D′n) is bad, and that GD′n =K1,n (cf. Figure 8).
Figure 8: The graph BC(D′3) (left) with each dotted line matching the two ‘halves’ of a
splitted vertex of D′3, and the graph GD′3 =K1,3 (right).
Similarly, each of the 2n nonempty components of PS(D′′n) is bad; this time it holds
that GD′′n = C2n (cf. Figure 9).
Interestingly, any simple graph G is a realization of some GD.
Proposition 2.7. For any simple graph G there exists a digraph D such that GD = G.
Proof. Let n = n(G) and m = m(G) be the order and size of G, respectively. An open
2m-necklace is a digraph obtained as follows: take a path P of length 2m, replace each
edge e ∈ E(P ) by a pair of parallel edges, and then orient each such pair consistently so
that with any natural ordering the vertices become: sink, source, sink, . . ., source, sink
(cf. Figure 10).
Take n disjoint open 2m-necklaces, and enumerate them as D1, . . . ,Dn. Consider an
enumeration of the set E(G) = {e1, . . . , em}. For each i ∈ {1, . . . , n}, fix a natural ordering
of the vertices of Di, and enumerate them accordingly as e0,i, e+1,i, e
−
1,i, . . . , e
+
m,i, e
−
m,i.
Let D be the digraph obtained from D1 ⊍ ⋯ ⊍Dn as follows. Take an enumeration of
the vertex set V (G) = {v1, . . . , vn}. For each ek ∈ E(G), if ek = vivj with i < j, then
identify e+k,i with e
−
k,j . Observe that, by construction:
• PS(D) has n nonempty components;
• each such component belongs to BC(D);
• GD = G.
The last item concludes our proof.
Recall that a matching in a graph is a set of pairwise nonadjacent edges that are not
loops. If M is a matching, the two ends of each edge of M are said to be matched under
M , and each vertex incident with an edge of M is said to be covered by M . A maximum
260 Ars Math. Contemp. 22 (2022) #P2.05 / 249–269
Figure 9: The graph BC(D′′6 ) (left) with each dotted line matching the two ‘halves’ of a
splitted vertex of D′′6 , and the graph GD′′6 = C12 (right).
Figure 10: The open 4-necklace.
matching in a given graph covers as many vertices as possible. The maximum matching
problem is the problem of finding a maximum matching in a given graph G. The num-
ber of edges in such a matching is called the matching number of G and denoted α′(G).
Thanks to the pioneering work of Tutte and Edmonds, the maximum matching problem is
known to be solvable in polynomial time. In particular, one of the 1965 papers of Edmonds
on polyhedral combinatorics, describes, among other things, the so-called Blossom Algo-
rithm [7] (see also [2], pp. 452)), an O(n2m) algorithm that finds a maximum matching
in any given graph of order n and size m. Over the years, various improvements of the
Blossom Algorithm have been found (see, e.g., [14], pp. 422–423).
Our next result establishes a relationship between the defect def(D) of any given di-
graph D and the order n(GD) and matching number α′(GD) of the corresponding simple
graph GD.
Theorem 2.8. For every digraph D, def(D) = n(GD) − α′(GD) holds.
Proof of Theorem 2.8. By Theorems 2.1 and 2.2, we may assume that χ′wo(D) = 3. Take
an arbitrary edge coloring φ of D with color set {1,2}. For simplicity, we use the same no-
tation φ to denote the inherited edge coloring of PS(D). Let PS(D)1 and PS(D)2 be the
spanning subgraphs of PS(D) whose respective edge sets are the color classes φ−1(1) and
φ−1(2). For every vertex x ∈ V (PS(D)), we abbreviate dPS(D)1(x) to d1(x), and likewise
dPS(D)2(x) to d2(x). Consider the partition {V (D)∩V (PS(D)), V (D)/V (PS(D))} of
V (D), and observe the following:
• a vertex u ∈ V (D) ∩ V (PS(D)) is ‘defective’ if and only if both d1(u) and d2(u)
C. Hernández-Cruz et al.: Notes on weak-odd edge colorings of digraphs 261
are odd;
• a vertex v ∈ V (D)/V (PS(D)) is ‘defective’ if and only if some v± ∈ {v+, v−} is a
nonisolated vertex of PS(D) such that both d1(v±) and d2(v±) are even (possibly
zero); call every such v± a ‘defective half-vertex’ originating from v.
First we show that each bad component of PS(D) ‘contains’ at least one defective
vertex.
Claim 2.8.1. Each component K of BC(D) contains a defective vertex or a defective
half-vertex.
Proof of Claim 2.8.1. Let K1 =K ∩PS(D)1 and K2 =K ∩PS(D)2. Since K is an even
graph, clearly OddV (K1) = OddV (K2) and EvenV (K1) = EvenV (K2). By the above
observations, within V (K), the defective vertices constitute the set V1 ∩OddV (K1) and
the defective half-vertices constitute the set V2 ∩ EvenV (K1). In order to show that the
union of these two sets is nonempty it suffices to note that
(V1 ∩OddV (K1)) ∪ (V2 ∩EvenV (K1)) = (V2 ∩ V (K))⊕OddV (K1) , (2.1)
the right-hand side being the symmetric difference of V2∩V (K) and OddV (K1). Now ob-
serve that V2∩V (K) is odd-sized by the assumption that K is bad. And, since OddV (K1)
is even-sized by the handshake lemma, we conclude that (V2 ∩ V (K)) ⊕ OddV (K1) is
odd-sized. Thus, the union of (2.1) is indeed nonempty, i.e., K contains a defective vertex
or a defective half-vertex.
We shall establish the desired equality def(D) = n(GD) − α′(GD) by showing that
each of the two opposed inequalities def(D) ≥ n(GD) −α′(GD) and def(D) ≤ n(GD) −
α′(GD) holds. In order to demonstrate the former inequality, we will need the following
auxiliary result.
Claim 2.8.2. Let G[X,Y ] be a simple bipartite graph such that for each vertex v ∈ X
the degree dG(v) is at most 2 and for each vertex w ∈ Y the degree dG(w) is positive. If
∣X ∣ =m and ∣Y ∣ = n then G contains at least n−m pairwise vertex-disjoint 2-paths whose
interior vertices belong to X .
Proof of Claim 2.8.2. Let p2(G) be the maximum size of a set of pairwise vertex-disjoint
2-paths in G with all interior vertices in X . We prove that p2(G) ≥ n −m by induction on
the number x2(G) of 2-vertices2 contained in X . If x2(G) = 0 then every vertex v ∈ X is
of degree at most 1. Since every vertex w ∈ Y is of degree at least 1, we have that
n −m = ∑
w∈Y
1 − ∑
v∈X
1 ≤ ∑
w∈Y
dG(w) − ∑
v∈X
dG(v) = 0 = p2(G) .
Assuming x2(G) ≥ 1, select a 2-vertex v0 ∈X . Define X ′ =X/{v0} and Y ′ = Y /NG(v0),
and let m′ = ∣X ′∣ and n′ = ∣Y ′∣; thus, m′ = m − 1 and n′ = n − 2. Note that the induced
subgraph G′[X ′, Y ′]meets the degree conditions. Since x2(G′) = x2(G)−1, the inductive
hypothesis gives n′ −m′ ≤ p2(G′). Therefore, as clearly p2(G′) ≤ p2(G) − 1, we deduce
that
n −m = (n′ −m′) + 1 ≤ p2(G′) + 1 ≤ p2(G) ,
which completes the inductive argument.
2A vertex v of a graph G is said to be a d-vertex if dG(x) = d.
262 Ars Math. Contemp. 22 (2022) #P2.05 / 249–269
We are ready to show one of the two opposed inequalities stated above.
Claim 2.8.3. def(D) ≥ n(GD) − α′(GD).
Proof of Claim 2.8.3. Returning to an arbitrary edge coloring φ of D with color set {1,2},
we construct a simple bipartite graph G[X,Y ] as follows. Let X be the set of defective
vertices in D under φ. Let Y be the set of components of BC(D). Join a defective vertex
v with a bad component K if K contains v or contains a defective half-vertex originating
from v. By Claim 1, the obtained graph G[X,Y ] meets the requirements of Claim 2.
Consequently, there are n(GD)− ∣X ∣ pairwise disjoint 2-paths in G[X,Y ] whose interiors
belong to X . However, this clearly gives a matching in GD of size n(GD) − ∣X ∣; simply
for every such 2-path y1xy2 from G[X,Y ] assign the edge vy1vy2 to the matching of GD.
We conclude that α′(GD) ≥ n(GD) − ∣X ∣. Equivalently, ∣X ∣ ≥ n(GD) − α′(GD). The
arbitrariness of φ yields the desired inequality.
In order to complete the proof of Theorem 2.8 we also need to prove the opposite
inequality.
Claim 2.8.4. def(D) ≤ n(GD) − α′(GD).
Proof of Claim 2.8.4. Consider a maximum matching M = {vK2i−1vK2i ∶ 1 ≤ i ≤ k} in
GD. Returning to PS(D), for each i ∈ {1, . . . , k} let vi ∈ V2 be a vertex such that {v+i , v−i }
intersects both K2i−1 and K2i. We color the edges of each nonempty component K of
PS(D) as described below. And for this, we define first an even-sized subset T of V (K):
• If K ∈ {K1,K2, . . . ,K2k} then define T = (V (K) ∩ V2)/{v+1 , v−1 , . . . , v+k , v−k}.
The intersection V (K) ∩ V2 is odd-sized and contains precisely one of the vertices
v+1 , v
−
1 , . . . , v
+
k , v
−
k ; hence, T is even-sized.
• If K is a component of BC(D) −⋃2ki=1Ki then there exists wK ∈ V2 such that the in-
tersection {w+K ,w−K}∩V (K) is a singleton; moreover, the ‘other half’ of {w+K ,w−K}
does not fall into another component of BC(D) −⋃2ki=1Ki, by the maximality of M .
Define T = (V (K) ∩ V2)/{w+K ,w−K}. Again, V (K) ∩ V2 is odd-sized and only one
of the vertices w+K ,w
−
K falls inside V (K) ∩ V2; consequently, T is even-sized.
• If K is not a component of BC(D) then (as in the proof of Theorem 2.2) we distin-
guish between two options: in case V (K)∩V2 is even-sized, define T = V (K)∩V2;
otherwise, as V (K) ∩ V3 ≠ ∅, select an odd-sized subset S ⊆ V (K) ∩ V3 and define
T = (V (K) ∩ V2) ∪ S. Obviously, T is even-sized.
By construction, T is always an even-sized subset of V (K). Therefore, there exists
a T -join H of K. Color E(H) with 1 and E(K)/E(H) with 2. After this has been
done for every component K of PS(D), consider the inherited edge coloring of D. The
set of its defective vertices is precisely R = {v1, . . . , vk} ∪ {wK ∶ K is a component of
BC(D) −⋃2ki=1Ki}. Indeed, by construction we have the following:
• no vertex u ∈ V (D) ∩ V (PS(D)) is defective because one of the colors 1 and 2
has an odd number of appearances on each of the odd-sized semi-cuts of u (as both
d1(u) and d2(u) are even);
C. Hernández-Cruz et al.: Notes on weak-odd edge colorings of digraphs 263
• a vertex v ∈ V (D)/V (PS(D)) is defective if and only if v ∈ R because those are the
only v’s for which some v± ∈ {v+, v−} is a nonisolated vertex of PS(D) such that
both d1(v±) and d2(v±) are even (possibly zero).
Thus, the total number of defective vertices equals n(GD) − α′(GD), which confirms the
desired inequality def(D) ≤ n(GD) − α′(GD).
Proof of Theorem 2.8, continued: From Claims 2.8.3 and 2.8.4 it follows that def(D) =
n(GD) − α′(GD).
Let us reconsider our examples. Since GD′n =K1,n and GD′′n = C2n, we have n(GD′n) =
n + 1 and α′(GD′n) = 1, whereas n(GD′′n) = 2n and α
′(GD′′n) = n; thus, in view of Theo-
rem 2.8, def(D′n) = def(D′′n) = n.
Note that Theorem 2.8 and Proposition 2.7 combined provide an answer to our previous
question about the complexity of finding the defect of a digraph.
Proposition 2.9. The parameter def(D) can be determined in polynomial time. Moreover,
the problem of finding the defect of a digraph is polynomially equivalent to the problem
finding the matching number of a graph.
Another immediate consequence of Theorem 2.8 is the following.
Corollary 2.10. For every digraph D it holds that
⌈n(GD)
2
⌉ ≤ def(D) ≤ n(GD) .
With all being said, it is clear that, in general, there is no ‘directed analogue’ of Propo-
sition 2.5, which served as our initial motivation here. In other words, there are digraphs
that have arbitrarily many ‘defective vertices’.
3 Characterizations in terms of χ′wo and def
We consider two classes of digraphs: the class AD = {D(G) ∶ G is a pseudograph} of
digraphs D(G) that are associated to graphs G, and the class T of tournaments.
3.1 Associated digraphs
We shall use here an additional convention hinted in the introduction. Namely, define
χ′wo(G) = ∞ for each graph G that contains ‘isolated loops’. The following theorem
characterizes the associated digraphs in terms of their weak-odd chromatic index.
Theorem 3.1. For any connected graph G, it holds that
χ′wo(D(G)) =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
0 if G =K1,
1 if G is an odd graph,
3 if G is an even bipartite graph of odd order ≥ 3,
2 otherwise.
264 Ars Math. Contemp. 22 (2022) #P2.05 / 249–269
. Let D =D(G). It always holds that
χ′wo(D) ≤ χ′wo(G) . (3.1)
Indeed, say φ is a weak-odd χ′wo(G)-edge coloring of G. The accompanying edge col-
oring φD of D assigns to any pair of arcs stemming from an edge e in G color φ(e).
Consequently, φD is a weak-odd χ′wo(G)-edge coloring of D(G), which settles (3.1).
Thus, the nontrivial part of Theorem 3.1 amounts to showing the following equivalence:
χ′wo(D) = 3 ⇔ G is an even bipartite graph of odd order n(G) ≥ 3 . (3.2)
In view of inequality (3.1) and Theorem 1.1, we may confine to G being an even graph of
odd order n(G) ≥ 3. With that assumption, clearly PS(D) = BG(D) and V1 = V3 = ∅.
Therefore, by Theorem 2.2, the equality χ′wo(D) = 3 holds true if and only if some
nonempty component of BG(D) is of odd order. Consequently, the proof of equiva-
lence (3.2) will be complete if we establish the following two assertions.
Claim 3.1.1. If G bipartite, then BG(D) = G ⊍ G, i.e., BG(D) consists of two vertex-
disjoint copies of G.
Claim 3.1.2. If G is not bipartite, then BG(D) is connected.
A moment’s reflection reveals that Claim 3.1.1 is implied by the definitions of ‘as-
sociated digraph’ and ‘split’. For if G = G[V1, V2] is a bipartite graph with bipartition
(V1, V2), then BG(D) = G[V +1 , V −2 ]⊍G[V −1 , V +2 ], that is, BG(D) is the disjoint union of
two bipartite graphs, with respective bipartitions (V +1 , V −2 ) and (V −1 , V +2 ), each of which
is isomorphic to G.
As for the demonstration of Claim 3.1.2, let x, y be an arbitrary pair of (not necessarily
distinct) vertices of G (and thus of D). In order to show the existence of an x+-y− walk
in BG(D), it suffices to find an x-y walk of odd length in G. Indeed, any such walk
W ∶ xv1v2⋯v2ky would yield a walk W ± ∶ x+v−1 v+2⋯v−2k−1v+2ky− in BG(D).
Consider an odd cycle C of G. Let P and Q, respectively, be an x-C and a y-C path in
G. Denote by vx and vy the (not necessarily distinct) endpoints of P and Q in C. Of the
two vxvy arcs of C, let L be the one whose length is of opposite parity than the combined
length ℓP + ℓQ of P and Q. Then P ∪L∪Q gives rise to a desired x-y walk of odd length.
A similar argument proves the existence of an x+-y+ walk in BG(D); it suffices to find
an x-y walk of even length in G, which can be done by using the other vxvy arc of C in the
previous argument. The existence of x−-y+ and x−-y− walks in BG(D) for an arbitrary
pair of vertices x and y in G now follows by symmetry.
An immediate consequence of Theorem 3.1 and inequality (3.1) is the following.
Corollary 3.2. If G is a connected graph, then χ′wo(D(G)) ≤ χ′wo(G). Moreover, equality
holds unless G is an even nonbipartite graph of odd order.
Let us characterize the associated digraphs in terms of their defect.
Proposition 3.3. For any connected graph G, it holds that
def(D(G)) =
⎧⎪⎪⎨⎪⎪⎩
1 if G is an even bipartite graph of odd order ≥ 3,
0 otherwise.
C. Hernández-Cruz et al.: Notes on weak-odd edge colorings of digraphs 265
Proof. By Theorem 3.1, unless G is an even bipartite graph of odd order n(G) ≥ 3, it holds
that def(D(G)) = 0. On the other hand, assuming G is an even bipartite graph of odd order
n(G) ≥ 3, by the proof of Theorem 3.1, PS(D) = BG(D) = G⊍G. Thus, BC(D) = G⊍G
and GD(G) =K2. Consequently, by Theorem 2.8, def(DG) = 1.
Taking into account the established inequality def(D(G)) ≤ 1, one naturally wonders if
an analogue of Proposition 2.5 holds for all associated digraphs. The following proposition
answers this in the positive.
Proposition 3.4. Every connected associated digraph D admits a 2-edge coloring such
that condition (
ÐÐ→
WO) is satisfied at each vertex apart from a prescribed vertex v ∈ V (D).
Proof. Let D = D(G). We may assume that G is an even bipartite graph of odd order
n(G) ≥ 3. As already observed in the proof of Proposition 3.3, it holds that PS(D) = G⊍G.
Let T = V (G)/{v}, and take a T -join H of G. Color the edges of PS(D) with color set
{1,2} as follows: in each copy of G, color E(H) by 1 and E(G)/E(H) by 2. The
inherited 2-edge coloring of D meets the requirements.
3.2 Tournaments
In view of Proposition 2.4, there exist tournaments that require three colors for a weak-odd
edge coloring; namely, as every tournament of odd order with a single peripheral vertex
meets the requirements of the aforementioned proposition, its weak-odd chromatic index
equals 3. Our characterization below asserts that those tournaments are the only ‘excep-
tions’ to weak-odd 2-edge colorability in the class T . The proof shall make use of the
following classical results on tournaments.
Given a digraph D, spanning directed paths and cycles are referred to as hamiltonian
paths and hamiltonian cycles, respectively. Back in 1959, Camion [3] proved that a non-
trivial tournament is strong if and only if it contains a hamiltonian cycle. (In fact, this basic
result was later on improved, first by Harary and Moser [8], and shortly after by Moon,
see, e.g., [9], but for our purposes the initial result of Camion will suffice.) Another basic
theorem on tournaments of an even earlier date, due to Rédei [13], is that every tourna-
ment (not necessarily strong) has a hamiltonian path. (In fact, Rédei [13] proved that every
tournament contains an odd number of hamiltonian paths.)
Theorem 3.5. For any tournament T , it holds that
χ′wo(T ) =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
0 if T =K1,
1 if T is nontrivial and every vertex semi-degree is odd or zero,
3 if T is nontrivial, of odd order, and has just one peripheral vertex,
2 otherwise.
Proof. For simplicity of presentation, call every nontrivial tournament of odd order having
only one peripheral vertex bad and call every other tournament good. By Proposition 2.4
and Theorem 2.1, the nontrivial aspect of the proof consists of showing that:
Every good tournament is weak-odd 2-edge colorable.
Consider a good tournament T . If it has two peripheral vertices, then the following
furnishes a weak-odd 2-edge coloring: take a hamiltonian path P of T , color A(P ) with
266 Ars Math. Contemp. 22 (2022) #P2.05 / 249–269
1 and the rest of A(T ) with 2. Since the initial (resp. terminal) vertex of P is the source
(resp. sink) of T , color 1 satisfies condition(
ÐÐ→
WO) at all vertices of T . Similarly, if every
strong component of T is nontrivial, then a simple construction of a weak-odd 2-edge
coloring can be obtained as follows: in every strong component K of T take a hamiltonian
cycle CK , color ⋃K A(CK) with 1 and the rest of A(T ) with 2. Then, color 1 meets
condition (
ÐÐ→
WO) everywhere.
Hence, we may assume that there exist a nontrivial peripheral strong component and
a trivial strong component of T . We complete the proof by distinguishing between two
cases.
Case 1: Both peripheral strong components of T are nontrivial. Let Ki and Kt be the
initial and terminal strong components of T . There exists a directed Ki-Kt path P in T
that passes through every vertex v ∉ V (Ki) ∪ V (Kj). Indeed, simply take a hamiltonian
path in the ‘multitournament’ T /{Ki,Kt}, i.e., the directed multigraph obtained from T
by contracting V (Ki) and V (Kt) into a pair of new vertices. By the above assumptions,
the path P is of length ℓ(P ) > 1. Denote by x and y, respectively, the initial and terminal
vertex of P . Thus, the arc xy ∈ A(T )/A(P ). Let Ci and Ct, respectively, be hamiltonian
cycles in Ki and Kj . The arc set A(Ci)∪A(P )∪{xy}∪A(Ct) induces a spanning subdi-
graph of T with all semi-degrees odd. By coloring A(Ci) ∪A(P ) ∪ {xy} ∪A(Ct) with 1
and the rest of A(T ) with 2 we complete a weak-odd 2-edge coloring of T , because color
1 meets condition (
ÐÐ→
WO) everywhere.
Case 2: One peripheral strong component of T is trivial. Since T is good, it has even order.
We may assume T has a sink, say y. Let Ki be the initial strong component of T , and Ci be
a hamiltonian cycle in Ki. If V (T ) = V (Ki) ∪ {y}, then we are done by coloring A(Ci)
with 1 and the rest of A(T ) with 2. Namely, color 2 meets condition (ÐÐ→WO) at y, and color
1 takes care of every other vertex.
Assuming V (T ) ≠ V (Ki) ∪ {y}, take a directed Ki-y path P in T that passes through
every vertex v ∉ V (Ki) (a hamiltonian path in the ‘multitournament’ T /Ki, the directed
multigraph obtained from T by contracting V (Ki) into a vertex, will do). Let x be the
initial vertex of P ; thus, V (Ci) ∩ V (P ) = {x}. By our latest assumption, the arc xy ∉
A(P ). Consequently, the arc set A(Ci) ∪A(P ) ∪ {xy} induces a spanning subdigraph of
T such that both the semi-degrees of y are even whereas the rest of the semi-degrees are
odd. Therefore, as dT (y) is odd, by coloring A(Ci) ∪A(P ) ∪ {xy} with 1 and the rest of
A(T ) with 2 we obtain a weak-odd 2-edge coloring of T . Indeed, once again color 2 meets
condition (
ÐÐ→
WO) at y, and color 1 takes care of every other vertex.
Let us characterize the members of the class T in terms of their defect.
Proposition 3.6. For any tournament T , it holds that
def(T ) =
⎧⎪⎪⎨⎪⎪⎩
1 if T is nontrivial, of odd order, and just one vertex semi-degree is zero,
0 otherwise.
Proof. By Theorem 3.5, we may assume that T is nontrivial, of odd order, and just one
vertex semi-degree is zero. Say T has a sink y. Apply to E(T ) the particular 2-edge
coloring(s) constructed for Case 2 in the proof of Theorem 3.5. Observe that condition
(ÐÐ→WO) is satisfied at each vertex apart from y.
C. Hernández-Cruz et al.: Notes on weak-odd edge colorings of digraphs 267
Therefore, as for the class of associated digraphs, the inequality def(T ) ≤ 1 holds for
every tournament T . Our final proposition shows that an analogue of Proposition 2.5 also
holds for all tournaments.
Proposition 3.7. Every tournament T admits a 2-edge coloring such that condition (
ÐÐ→
WO)
is satisfied at each vertex apart from a prescribed vertex v ∈ V (T ).
Proof. Again, we may assume that T is nontrivial, of odd order, and with just one periph-
eral vertex, say a sink y. Note that BG(T ) has only two components, and moreover, one of
those components consists of the isolated vertex y+. Indeed, by our assumptions, every ver-
tex w ∈ V (T )/{y} dominates y and has d−T (w) > 0; hence, the component containing y−
also includes both w+ and w−. Consequently, PS(T ) has only one nonempty component
K and only one empty component {y+}. Observe that V (K)∩V2 = V2/{y+} is odd-sized,
and V3 = ∅. Define an even-sized subset S ⊆ V (K) as follows:
• if v ∈ V1 then S = {v} ∪ (V2/{y+});
• if v ∉ V1 then S = V2/{v−, y+}.
The rest should be clear. We simply take an S-join H of K, and then color E(H) with 1
and E(K)/E(H) with 2. The inherited 2-edge coloring of T meets the requirements.
4 Concluding remarks and further work
For a graph G (resp. digraph D), an edge covering with color set S is a mapping that assigns
to each edge of G (resp. arc of D) a nonempty subset of S; what distinguishes coverings
from colorings is that we allow more than one color per edge (resp. arc). Related notions
to weak-odd edge colorings of graphs and digraphs, respectively, are the weak-odd edge
coverings defined as edge coverings such that conditions (WO) and (
ÐÐ→
WO) are fulfilled. It
is known that most of the graphs and digraphs are weak-odd 3-edge colorable. Can a color
always be saved by switching to coverings? The answer to this question in the realm of
graphs is affirmative. Indeed, the following holds true.
Proposition 4.1. Any connected graph G whose edge set does not consist entirely of loops,
admits a weak-odd 2-edge covering such that the intersection of color classes is contained
within a prescribed singleton {e} ⊆ E(G).
Proof. By Theorem 1.1, we may assume that G is a nontrivial even graph of odd order.
Subdivide the edge e, and take an odd factor H of the obtained graph. Color E(G)∩E(H)
with 1, E(G)/(E(H) ∪ {e}) with 2, and assign both colors 1 and 2 to the edge e. It is
readily seen that the constructed edge covering meets the requirements.
Following this line of reasoning, we find the next question interesting.
Question 4.2. Does every digraph admit a weak-odd 2-edge covering?
Presuming Question 4.2 answers in positive, define ovl(D), the overlapping of D, to
be the minimum possible size of the intersection of the two color classes in an arbitrary
weak-odd 2-edge covering of D. In view of the families of digraphs D′n and D
′′
n (de-
picted in Figure 7), it is easily seen that ovl(D) is not bounded over the class of digraphs;
moreover, it can acquire any possible value from the set of naturals. We are tempted to
268 Ars Math. Contemp. 22 (2022) #P2.05 / 249–269
wonder whether this parameter also relates to some ‘classical graph parameter’, much as
like def(D) relates to the maximum matching number of graphs.
Following the direction explored in Section 3, it may be interesting to characterize
other digraph families in terms of their weak-odd chromatic index and their defect. Since
tournaments proved to have a nice behavior with respect these parameters, a natural next
step is to consider families of digraphs generalizing tournaments.
Three classic generalizations of tournaments that come to mind are semicomplete di-
graphs, extended tournaments and multipartite tournaments. A digraph is semicomplete
if it is obtained from a complete graph by replacing each edge uv by the arc (u, v), the
arc (v, u) or the pair of arcs (u, v) and (v, u). An extended tournament is a digraph ob-
tained from a tournament by blowing up some of its vertices into stable sets. A multipartite
tournament is an orientation of a complete multipartite graph.
Problem 4.3. Characterize the families of semicomplete digraphs, extended tournaments
and multipartite tournaments in terms of their weak-odd chromatic index.
We think that the following question should be addressed before stating the analogous
problem for the characterization in terms of the defect.
Question 4.4. Is there a constant c such that def(D) ≤ c for every digraph D such that
• D is semicomplete?
• D is an extended tournament?
• D is a multipartite tournament?
A positive answer for Question 4.4 would open the door to consider the following
problem.
Problem 4.5. Characterize the families of semicomplete digraphs, extended tournaments
and multipartite tournaments in terms of their defect.
ORCID iDs
César Hernández-Cruz https://orcid.org/0000-0002-5867-3801
Mirko Petruševski https://orcid.org/0000-0001-8048-7418
Riste Škrekovski https://orcid.org/0000-0001-6851-3214
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.06 / 271–286
https://doi.org/10.26493/1855-3974.2053.c7b
(Also available at http://amc-journal.eu)
A-trails of embedded graphs and twisted duals*
Qi Yan †
School of Mathematics, China University of Mining and Technology,
Xuzhou 221116, P. R. China, and
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China
Xian’an Jin ‡
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China
Received 18 July 2019, accepted 15 August 2021, published online 27 May 2022
Abstract
Kotzig showed that every connected 4-regular plane graph has an A-trail—an Eulerian
circuit that turns either left or right at each vertex. However, this statement is not true for
Eulerian plane graphs and determining if an Eulerian plane graph has an A-trail is NP-
hard. The aim of this paper is to give a characterization of Eulerian embedded graphs
having an A-trail. Andersen et al. showed the existence of orthogonal pairs of A-trails
in checkerboard colourable 4-regular graphs embedded on the plane, torus and projective
plane. A problem posed in their paper is to characterize Eulerian embedded graphs (not
necessarily checkerboard colourable) which contain two orthogonal A-trails. In this article,
we solve this problem in terms of twisted duals. Several related results are also obtained.
Keywords: Embedded graphs, twisted duals, Eulerian, A-trails, checkerboard colourable.
Math. Subj. Class. (2020): 05C10, 05C45, 05C62
1 Introduction
A cellularly embedded graph is a graph G embedded in a surface Σ such that every con-
nected component of Σ − G is a 2-cell, called a face. We use the term embedded graph
loosely to mean any of three equivalent representations of graphs in surfaces: cellularly em-
bedded graphs, ribbon graphs and arrow presentations. We shall move from one to another
*We thank Graham Farr, Yichao Chen and the referees sincerely for their valuable comments.
†Corresponding author. This work is supported by NSFC (No. 12101600) and the Fundamental Research
Funds for the Central Universities (No. 2021QN1037)
‡This work is supported by NSFC (No. 12171402) and the Fundamental Research Funds for the Central Uni-
versities (No. 20720190062).
E-mail addresses: qiyan@cumt.edu.cn (Qi Yan), xajin@xmu.edu.cn (Xian’an Jin)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
272 Ars Math. Contemp. 22 (2022) #P2.06 / 271–286
freely and refer the reader to [5, 6, 10] for details. A quasi-tree is an embedded graph with
exactly one boundary component (or face). A bouquet is an embedded graph with exactly
one vertex. They are geometric duals to each other. A quasi-tree bouquet is a bouquet that
is also a quasi-tree.
In this article, all graphs will be finite, connected, but not necessarily simple. A graph
is said to be Eulerian if the degree of each of its vertices is even. A graph is bipartite if its
vertex set can be partitioned into two nonempty subsets X and Y so that every edge has
one end in X and one end in Y . We denote a bipartite graph G with bipartition (X,Y )
by G[X,Y ]. Note that bipartite graphs might have multiple edges but not loops. A star
is a bipartite graph G[X,Y ] with |X| = 1 or |Y | = 1. A parallel graph, also known
as a generalized theta graph especially for 3 or more edges, is a special star G[X,Y ] with
|X| = 1 and |Y | = 1. A walk in a graph between vertices v0 and vk is a sequence of
vertices and edges (v0, e1, v1, e2, · · · , ek, vk), where vi−1 and vi are the endvertices of the
edge ei. A trail is a walk with no repeated edge and circuit is a trail with v0 = vk. An
A-trail [7] in an Eulerian embedded graph G is an eulerian circuit such that every two
consecutive edges in the circuit are adjacent in the rotation of the common vertex. A Petrie
walk in an embedded graph G is such a walk that when traveling along it, we alternatingly
turn to the left edge and to the right edge of the current edge in the cyclic rotation around
the common vertex. It is obvious that an Eulerian Petrie walk is a special kind of A-trails.
Kotzig [7] showed that every 4-regular plane graph has an A-trail, and sufficient con-
ditions were discovered for the existence of A-trails in 2-connected plane graphs in [2].
The existence of A-trails has been studied almost exclusively in the case of graphs em-
bedded in the plane, the projective plane and the torus. In this paper, we shall charac-
terize general Eulerian embedded graphs having an A-trail in terms of twisted duals. Let
G = (V (G), E(G)) be an embedded graph. We denote by G∗ and G× the geometric
dual and Petrial [14] of G, respectively. Let A ⊆ E(G). We denote by Gδ(A) and Gτ(A)
the partial dual [4] and partial Petrial of G with respect to A, respectively. Particularly, if
A = E(G), then Gδ(E(G)) = G∗ and Gτ(E(G)) = G×. Partial duality and partial Petriality
are further combined together to form twisted duality [1, 5]. The twisted duality has the
scope to develop the understanding of a wide variety of graph theoretical problems. We
refer the reader to [5, 6] for the details. Petrie walks have some very interesting properties,
see [8, 12]. They also play an important role in the design of CMOS VLSI circuits, where
it is convenient if the graph representing a circuit has an Eulerian Petrie walk. Žitnik [13]
gave a characterization of 4-regular plane graphs with Eulerian Petrie walks. Since Eule-
rian Petrie walks are a special kind of A-trails, we also give a characterization of Eulerian
embedded graphs having an Eulerian Petrie walk. We first obtain the following theorem.
Theorem 1.1. Let G be an Eulerian embedded graph. Then G has an A-trail if and only if
there exists B ⊆ E(G) such that the underlying graph of (Gτ(B))∗ is a star. In particular,
G has an Eulerian Petrie walk if and only if the underlying graph of (G×)∗ is a star.
Andersen, Bouchet and Jackson [3] characterized the 4-regular plane graphs which
contain two orthogonal A-trails, that is to say two A-trails for which no subtrail of length 2
appears in both A-trails. They also discussed the corresponding problem for checkerboard
colourable graphs embedded in the projective plane and the torus. And they posed the
following problem.
Problem 1.2 ([3]). We do not have any characterization of 4-regular graphs in the projec-
tive plane and the torus having two orthogonal A-trails which we know to be valid also for
Q. Yan and X. Jin: A-trails of embedded graphs and twisted duals 273
graphs with no 2-face colouring.
In this paper, we give a general definition of two orthogonal A-trails in Eulerian em-
bedded graphs. We say that two A-trails are orthogonal if these two trails have different
transitions at each vertex with degree greater than 2. We consider the above problem for
general Eulerian embedded graphs which may have high genus and are not necessarily
checkerboard colourable and characterize the Eulerian embedded graphs which have two
orthogonal A-trails or Eulerian Petrie walks in terms of twisted duals as follows.
Theorem 1.3. Let G be an embedded graph. Then G has two orthogonal A-trails if and
only if there exists B ⊆ E(G) such that the underlying graph of (Gτ(B))∗ is a parallel
graph. In particular, G has two orthogonal Eulerian Petrie walks if and only if the under-
lying graph of (G×)∗ is a parallel graph.
In the case of 4-regular embedded graphs, we have the following theorem.
Theorem 1.4. Let H be a 4-regular embedded graph. Then H has two orthogonal A-trails
if and only if there exists D ⊆ E(H) such that Hτ(D) is a medial graph of a quasi-tree
bouquet. Particularly, H has two orthogonal Eulerian Petrie walks if and only if H× is a
medial graph of a quasi-tree bouquet.
2 Preliminaries
In this section we introduce the notions and the tools, which we will need in further Sec-
tions 3 and 4. We use standard notations V (G), E(G) and F (G) to denote the sets of ver-
tices, edges, and faces, respectively, of a cellularly embedded graph G and
v(G) = |V (G)|, e(G) = |E(G)| and f(G) = |F (G)|, respectively. We denote by d(v) the
degree of a vertex v in G, i.e. the number of half-edges incident with v. We give a brief
review of ribbon graphs referring the reader to [5, 6] for further details.
Definition 2.1 ([6]). A ribbon graph G = (V (G), E(G)) is a (possibly non-orientable)
surface with boundary, represented as the union of two sets of topological discs, a set
V (G) of vertices, and a set E(G) of edges such that
1. the vertices and edges intersect in disjoint line segments, we call them common line
segments as in [9];
2. each such common line segment lies on the boundary of precisely one vertex and
precisely one edge;
3. every edge contains exactly two such common line segments.
Let G be a ribbon graph, v ∈ V (G) and e ∈ E(G). By deleting the common line
segments from the boundary of v, we obtain d(v) disjoint line segments, called vertex
line segments. By deleting common line segments from the boundary of e, we obtain two
disjoint line segments, called edge line segments. See Figure 1 for an example of these
concepts. We think of each edge line segment having two half-edge line segments. It is
obvious that every edge disc contains four half-edge line segments which correspond to
the four flags incident on that edge. For any vertex line segment, there are exactly two
half-edge line segments incident with it as shown in Figure 2.
274 Ars Math. Contemp. 22 (2022) #P2.06 / 271–286
Figure 1: Vertex line segments (yellow), common line segments (red) and edge line seg-
ments (blue).
Let G be a ribbon graph and A ⊆ E(G). Then the partial Petrial, Gτ(A), of G with
respect to A is the ribbon graph obtained from G by adding a half-twist to each of the edges
in A. Let Orb(τ)(G) = {Gτ(A)|A ⊆ E(G)} denote the set of all partial Petrials of G. Let
H be an arrow presentation and B ⊆ E(H). Then the partial dual, Hδ(B), of H with
respect to B is the arrow presentation obtained as follows. For each e ∈ B, suppose α and
β are the two arrows labelled e in the arrow presentation of H . Draw a line segment with
an arrow on it directed from the head of α to the tail of β, and a line segment with an arrow
on it directed from the head of β to the tail of α. Label both of these arrows e and delete α
and β and the arcs containing them. This process is illustrated locally at a pair of arrows in
Figure 3. Let Orb(δ)(H) = {Hδ(B)|B ⊆ E(H)} denote the set of all partial duals of H .
Figure 2: Four half-edge line segments of e, one of the vertex line segments of v and its
incident two half-edge line segments.
Let B =< δ, τ |δ2, τ2, (δτ)3 >.
Definition 2.2 ([5]). Let G be a ribbon graph. The ribbon graph H is called a twisted dual
(briefly, twual) of G if it can written in the form
H = GΠ
6
i=1ξi(Ai),
where the Ai’s partition E(G) and the ξi’s are the six elements of B.
Q. Yan and X. Jin: A-trails of embedded graphs and twisted duals 275
Figure 3: Taking the partial dual of an edge in an arrow presentation.
If G is cellularly embedded in Σ, we construct its medial graph Gm in the embed-
ded surface by placing a vertex on each of its edges, and for each face f with boundary
e1, e2, · · · , ed(f), drawing d(f) edges {e1, e2}, · · · , {ed(f), e1} inside the face f along the
boundary of f . It is obvious that Gm is also cellularly embedded in Σ. In particular, the
medial graph of an isolated vertex is a free loop. G is checkerboard colourable if the faces
of G can be properly 2-coloured. A checkerboard colouring of G is a particular proper
2-colouring of the faces of G. Throughout we will use red and blue to refer to the two
colours used in a checkerboard colouring. It is obvious that there is a correspondence be-
tween checkerboard colouring and face boundary colouring, so we shall move from one
to another freely. Note that in a checkerboard coloured 4-regular embedded graph G, the
redface graph GR of G is the embedded graph constructed by placing one vertex in each
red face and adding an edge between two of these vertices whenever the corresponding
faces meet at a vertex of G. The blueface graph GB is constructed analogously by placing
vertices in the blue faces.
A Petrie walk in an embedded graph G is such a walk that when traveling along it, we
alternatingly turn to the left edge and to the right edge of the current edge in the cyclic rota-
tion around the common vertex. We shall only consider closed Petrie walks, for which this
condition holds also for the last and the first edge of the walk. Petrie walks are sometimes
also called left-right paths. An example of a Petrie walk is shown in Figure 4, where the
dotted curve indicates the order of edges in the walk. Note that the boundary components
of faces of the Petrie dual are exactly Petrie walks of the original embedding of G as shown
in Figure 5.
Figure 4: The curve representing a Petrie walk.
Let v be a vertex of G. A transition at v is a partition of the half-edges incident to v into
276 Ars Math. Contemp. 22 (2022) #P2.06 / 271–286
Figure 5: A Petrie walk corresponding to a face of the Petrie dual of the original embedded
graph.
pairs. A transition T at v is smooth if T only pairs half-edges adjacent in the cyclic order at
v given by the embedding of G. A transition system of G is a choice of a transition at every
vertex of G. A smooth transition system is a transition system such that every transition
is smooth. It is obvious that we can induce a circuit decomposition of G by a transition
system T . Similarly, any circuit decomposition C of G recovers a transition system of G
by pairing half-edges traversed consecutively in a circuit of C. A circuit decomposition of
G is smooth if it induces a smooth transition system. In particular, an Eulerian circuit that
induces a smooth transition system is called an A-trail [11]. Note that there are precisely
two disjoint smooth transitions for any vertex v of G with d(v) ≥ 4 and two edges are
consecutive if this is indicated by a curve as shown in Figure 6. We say that two A-trails
Figure 6: Performing exactly two smooth transitions locally at a vertex.
(or smooth transition systems) of an Eulerian embedded graph G are orthogonal if the two
trails (or smooth transition systems) have different smooth transitions at each vertex v of G
with d(v) ≥ 4.
Most of the representations of embedded graphs are ribbon graphs in this paper, so we
introduce the relation between smooth transition systems and ribbon graphs. Let G be an
Eulerian ribbon graph. For every v ∈ V (G), we assign the colours red and blue to all
half-edge line segments and vertex line segments of v such that the colours of vertex line
segments are alternating red and blue in the vertex boundary of v and every vertex line
segment and two half-edge line segments incident with it assign the same colour. We call
this a checkerboard colouring of half-edge and vertex line segments of G (see Figure 7 for
Q. Yan and X. Jin: A-trails of embedded graphs and twisted duals 277
example). If G is checkerboard coloured of half-edge and vertex line segments, then for any
edge disc, there are exactly two cases as shown in Figure 8. The edge is called consistent
if two half-edge line segments of one of edge line segments have the same colour, and is
called inconsistent otherwise. Let T be a smooth transition system of G. It is obvious that
T induces a checkerboard colouring of half-edge and vertex line segments for any vertex of
G as shown in Figure 9. We call this a canonical checkerboard colouring of half-edge and
vertex line segments by T . An example of a canonical checkerboard colouring of half-edge
and vertex line segments by the smooth transition system is given in Figure 10.
Figure 7: A checkerboard colouring of half-edge and vertex line segments at a vertex.
Figure 8: (a) consistent edge (b) inconsistent edge.
3 Main results and proofs
Now we give some characterizations of Eulerian embedded graphs having an A-trail or
Eulerian Petrie walk. Kotzig [7] showed that every 4-regular plane graph has an A-trail.
We note that this result can be extended to any 4-regular embedded graph.
Lemma 3.1. Any 4-regular embedded graph always has an A-trail.
Proof. Let T be any smooth transition system of a 4-regular embedded graph H . Note
that each smooth transition system corresponds to a specific family of edge-disjoint cycles
278 Ars Math. Contemp. 22 (2022) #P2.06 / 271–286
Figure 9: The relation between smooth transition and checkerboard colouring of half-edge
and vertex line segments at a vertex.
Figure 10: An example of a canonical checkerboard colouring of half-edge and vertex line
segments by the smooth transition system.
Q. Yan and X. Jin: A-trails of embedded graphs and twisted duals 279
in H . The number of edge-disjoint cycles in H generated by T will be denoted by c(T ).
If c(T ) = 1, then this completes the proof. Otherwise, there exists v ∈ V (H) such
that four half-edges 1, 2, 3, 4 incident with v are not in the same cycle. Assume that the
transition at v is (1, 2), (3, 4). Then we change the smooth transition of v from (1, 2), (3, 4)
to (1, 4), (2, 3). Hence, half-edges 1, 2, 3, 4 are in the same edge-disjoint cycle as shown
in Figure 11. Repeating the above process, we obtain a smooth transition system which
corresponds to an A-trail.
Figure 11: Proof of Lemma 3.1.
Lemma 3.1 can not be further generalized to any Eulerian embedded graph and an
example is given in Figure 12. Note that taking partial petrial does not affect cyclic order
Figure 12: An Eulerian embedded graph which does not have an A-trail.
of half-edges at vertices, we have the following lemma.
Lemma 3.2. Let G be an Eulerian embedded graph. Then
(1) G has an A-trail if and only if every partial Petrial of G has an A-trail.
(2) G has two orthogonal A-trails if and only if every partial Petrial of G has two or-
thogonal A-trails.
Theorem 1.1. Let G be an Eulerian embedded graph. Then G has an A-trail if and only if
there exists B ⊆ E(G) such that the underlying graph of (Gτ(B))∗ is a star. In particular,
G has an Eulerian Petrie walk if and only if the underlying graph of (G×)∗ is a star.
280 Ars Math. Contemp. 22 (2022) #P2.06 / 271–286
Proof. (⇒) Let G be a ribbon graph. If G has an A-trail, then there exists a smooth
transition system T of G corresponding to the A-trail. We get a canonical checkerboard
colouring of half-edge and vertex line segments according to T . Suppose that B is the set
of its inconsistent edges. If the A-trail is an Eulerian Petrie walk, then B = E(G). Thus
Gτ(B) obtains a face boundary colouring. It induces a checkerboard colouring with only
one red face. Then (Gτ(B))∗ is bipartite with |X| = 1 or |Y | = 1, hence the underlying
graph of (Gτ(B))∗ is a star.
(⇐) Since the underlying graph of (Gτ(B))∗[X,Y ] is a star, we assume that X = {v}.
Note that v corresponds to a face of Gτ(B). We assign the colour red to this face and colour
blue to other faces of Gτ(B). This is a checkerboard colouring of Gτ(B). Suppose that it
is a canonical checkerboard colouring of half-edge and vertex line segments of Gτ(B). Let
T be the corresponding smooth transition system. It follows that T induces an A-trail of
Gτ(B). Thus, G has an A-trail by Lemma 3.2. If B = E(G), then this is a checkerboard
colouring of G×. Hence, the boundary of the redface of G× is an Eulerian Petrie walk of
G.
Remark 3.3. According to Theorem 1.1, suppose that G is an embedded graph whose
underlying graph is a star. If H ∈ Orb(τ)(G∗), then H has an A-trail. Particularly, if
H = (G∗)×, then H has an Eulerian Petrie walk.
Corollary 3.4. Let H be a 4-regular embedded graph. Then H has an Eulerian Petrie
walk if and only if H× is a medial graph of a bouquet.
Proof. By a similar argument as in the proof of Theorem 1.1, H× can obtain a checker-
board colouring with only one red face. Hence, the number of vertices of the redface graph
(H×)R is exactly one, that is a bouquet. Therefore, H× is a medial graph of a bouquet.
Conversely, let G be a bouquet and H× is the medial graph of G. We give Gm a checker-
board colouring where the red faces contain the vertices of G. Then the number of red
faces is exactly one. Hence, (Gm)∗ is a star. Thus, (Gm)× has an Eulerian Petrie walk by
Remark 3.3. Since Gm = H×, we can see that H has an Eulerian Petrie walk.
Theorem 1.3. Let G be an embedded graph. Then G has two orthogonal A-trails if and
only if there exists B ⊆ E(G) such that the underlying graph of (Gτ(B))∗ is a parallel
graph. In particular, G has two orthogonal Eulerian Petrie walks if and only if the under-
lying graph of (G×)∗ is a parallel graph.
Proof. (⇒) Assume that T and T ′ are two orthogonal smooth transition systems recov-
ering from the two orthogonal A-trails of a ribbon graph G. Then we get a canonical
checkerboard colouring of half-edge and vertex line segments according to T . Suppose
that B is the set of its inconsistent edges. If the A-trail is an Eulerian Petrie walk, then
B = E(G). By the same argument as in the proof of Theorem 1.1, there is a face boundary
colouring of Gτ(B) such that the number of colour red face boundaries is exactly one. Note
that T ′ corresponds to the blue face boundaries of Gτ(B). It follows that the number of
colour blue face boundary is also exactly one. Hence, the vertex set of (Gτ(B))∗ can be
partitioned into two subsets X and Y with |X| = |Y | = 1. Thus, the underlying graph of
(Gτ(B))∗ is a parallel graph.
(⇐) Since the underlying graph of (Gτ(B))∗ is a parallel graph with vertices v and w.
Note that the number of face boundaries of Gτ(B) are two, which correspond to v and w,
respectively. We give one colour red and another colour blue. Hence, this is a checkerboard
Q. Yan and X. Jin: A-trails of embedded graphs and twisted duals 281
colouring of Gτ(B). Let T and T ′ be the corresponding two orthogonal smooth transition
systems to the red face boundary and the blue face boundary, respectively. It is obvious that
T and T ′ induce two orthogonal A-trails of Gτ(B). Thus, G has two orthogonal A-trails
by Lemma 3.2. If B = E(G), then this is a checkerboard colouring of G×. Note that
the redface and blueface of G× are both Eulerian Petrie walks of G. Hence, G has two
orthogonal Eulerian Petrie walks.
Remark 3.5. According to Theorem 1.3, suppose that G is an embedded graph whose
underlying graph is a parallel graph. If H ∈ Orb(τ)(G∗), then H has two orthogonal
A-trails. In particular, if H = (G∗)×, then H has two orthogonal Eulerian Petrie walks.
Theorem 1.4. Let H be a 4-regular embedded graph. Then H has two orthogonal A-trails
if and only if there exists D ⊆ E(H) such that Hτ(D) is a medial graph of a quasi-tree
bouquet. Particularly, H has two orthogonal Eulerian Petrie walks if and only if H× is a
medial graph of a quasi-tree bouquet.
Proof. (⇒) By the same argument as in the proof of Theorem 1.3, there exists D ⊆ E(H)
such that there is a checkerboard colouring of Hτ(D) which the number of colour red
face and blue face are both exactly one, that is, the number of vertices of (Hτ(D))R and
(Hτ(D))B are both exactly one. Note the number of face of (Hτ(D))R is also one, since
(Hτ(D))R and (Hτ(D))B are geometric duals. Therefore, (Hτ(D))R is a quasi-tree bou-
quet. Hence, Hτ(D) is a medial graph of a quasi-tree bouquet.
(⇐) Let G be a quasi-tree bouquet and Hτ(D) be the medial graph of G. Since G and
Gm embedded in the same surface, we have v(G) − e(G) + f(G) = v(Gm) − e(Gm) +
f(Gm) by Euler characteristic. Note that v(Gm) = e(G), e(Gm) = 2v(Gm) = 2e(G).
Hence, f(Gm) = v(G)+f(G) = 2. Thus, (Gm)∗ is a parallel graph since Gm is checker-
board colourable and f(Gm) = 2. Then Gm has two orthogonal A-trails by Theorem 1.3,
that is, Hτ(D) has two orthogonal A-trails. Hence, H has two orthogonal A-trails by
Lemma 3.2. If D = E(H), then (H×)∗ is a parallel graph. It follows that H has two
orthogonal Eulerian Petrie walks by Remark 3.5.
Remark 3.6. According to Theorem 1.4, suppose that G is a quasi-tree bouquet. If H ∈
Orb(τ)(Gm), then H has two orthogonal A-trails. In particular, if H = (Gm)×, then H
has two orthogonal Eulerian Petrie walks.
Lemma 3.7. Let G be an embedded graph. Then E(G) can be partitioned into two edge
disjoint spanning quasi-trees if and only if G is the partial dual of a quasi-tree bouquet.
Proof. Suppose A and Ac are the edge sets of two spanning quasi-trees which partition
E(G), then Gδ(A) is a bouquet since the vertex boundaries of Gδ(A) correspond to the face
boundaries of (V (G), A) which is a spanning quasi-tree. Thus, Gδ(A
c) is also a bouquet by
the similar discussion. Note that Gδ(A) = (Gδ(A
c))∗. Hence, Gδ(A) is a quasi-tree bouquet,
that is, G is the partial dual of a quasi-tree bouquet. Conversely, there exists A ⊆ E(G)
such that Gδ(A) is a quasi-tree bouquet. Then Gδ(A
c) is also a bouquet. It follows that
(V (G), A) and (V (G), Ac) are both spanning quasi-trees, that is, E(G) can be partitioned
into two edge disjoint spanning quasi-trees.
Lemma 3.8 ([5]). Let G be an embedded graph. Then
Orb(δ)(G) = {H|Gm and Hm are partial Petrials}.
282 Ars Math. Contemp. 22 (2022) #P2.06 / 271–286
Corollary 3.9. Let H be a checkerboard coloured 4-regular embedded graph. Then H has
two orthogonal A-trails if and only if the edges of the redface graph HR can be partitioned
into two edge disjoint spanning quasi-trees.
Proof. E(HR) can be partitioned into two edge disjoint spanning quasi-trees if and only if
HR is the partial dual of a quasi-tree bouquet by Lemma 3.7, that is, there exists D ⊆ E(H)
such that Hτ(D) is a medial graph of a quasi-tree bouquet by Lemma 3.8, if and only if H
has two orthogonal A-trails by Theorem 1.4.
Corollary 3.10. Let H be a checkerboard coloured 4-regular orientable embedded graph
which has two orthogonal A-trails. Then v(H) is even.
Proof. E(HR) can be partitioned into two edge disjoint spanning quasi-trees by Corol-
lary 3.9. Denote these two spanning quasi-trees by G1 = (V (HR), E1), G2 = (V (HR), E2),
respectively. Obviously, G1 and G2 are both orientable. Then v(HR) − |E1| + 1 =
2− 2g(G1) and v(HR)− |E2|+ 1 = 2− 2g(G2), where g(G1) and g(G2) are the genera
of G1 and G2, respectively. Hence, |E1| + |E2| = 2v(HR) + 2g(G1) + 2g(G2) − 2. It
follows that e(HR) is even, that is, v(H) is even.
Remark 3.11. Andersen, Bouchet and Jackson [3] obtained the same results as Corollar-
ies 3.9 and 3.10 for graphs embedded in the plane, the projective plane and the torus.
4 Quasi-tree bouquets
Theorem 1.4 and Lemma 3.7 show that quasi-tree bouquets are an important class of ribbon
graphs. In this section, we give a brief characterization of them.
We start by recalling some necessary statements. A ribbon graph is non-orientable if
it contains a ribbon subgraph that is homeomorphic to a Möbius band, and is orientable
otherwise. An edge e of a ribbon graph is a loop if it is incident with exactly one vertex.
A loop is non-orientable if together with its incident vertex it forms a Möbius band, and
is orientable otherwise. Two loops e and f are interlaced if they are met in the cyclic
order efef when travelling round the boundary of a vertex. A loop e at the vertex of a
bouquet G is trivial if there is no loop in G which interlaces with e. The signed rotation of
a bouquet is a cyclic ordering of the half-edges at the vertex and if the edge is orientable,
then we give the same sign to the corresponding two half-edges, and give the different signs
otherwise. See Figure 13 for an example. Suppose P = p1p2 · · · pk is a string. Then we
call −P = (−pk) · · · (−p2)(−p1) the inverse of P . Operations 1 and 2 are the moves on
bouquets defined in Figure 14 and Figure 15, respectively. Operation 3 is deleting a pair
of interlaced orientable loops and operation 4 is deleting a trivial non-orientable loop as
shown in Figure 16 and Figure 17, respectively. Note that operations 1, 2, 3 and 4 do not
change the number of boundary components.
Theorem 4.1. Let G be a bouquet. Then G is a quasi-tree bouquet if and only if there is a
sequence of operations 1, 2, 3 and 4 which change the signed rotation of G to be empty.
Proof. The sufficiency is easily verified. To prove the necessity, the result is easily verified
when E(G) = ∅, so assume that this is not the case. Then there are following two cases.
Case 1: If there exists a non-orientable loop r, we assume that the signed rotation of G is
IrJ(−r)P. By a sequence of operation 2, we have I(−J)r(−r)P . Hence, we get a signed
rotation I(−J)P by operation 4.
Q. Yan and X. Jin: A-trails of embedded graphs and twisted duals 283
Figure 13: The signed rotation of the bouquet is ab(−a)cb(−c)dd.
Figure 14: Operation 1. Change the signed rotation from AaBba to AabBa.
Figure 15: Operation 2. Change the signed rotation from AaBb(−a) to A(−b)aB(−a).
284 Ars Math. Contemp. 22 (2022) #P2.06 / 271–286
Figure 16: Operation 3. Change the signed rotation from Aabab to A.
Figure 17: Operation 4. Change the signed rotation from Aa(−a) to A.
Q. Yan and X. Jin: A-trails of embedded graphs and twisted duals 285
Case 2: Otherwise, there exists a pair of interlaced orientable loops e, f . Assume that the
signed rotation of G is IeJfPeQfM. By a sequence of operation 1, we have
IeJfPeQfM ⇔ IefPeQfJM ⇔ IefeQPfJM ⇔ IQPefefJM.
Then by operation 3, we get a signed rotation IQPJM.
Note that both Case 1 and Case 2 induce a shorter signed rotation. By repeating the
above operations, we can reduce the signed rotation of G until it is empty.
Remark 4.2. A bouquet with signed rotation AxyB(−x)(−y)C is not a quasi-tree bou-
quet since AxyB(−x)(−y)C ⇔ Axx(−B)y(−y)C.
We now present an algorithm to get the number of boundary components of any bouquet
G in terms of signed rotations.
Algorithm 1 Calculate the number of boundary components of any bouquet.
1: Input The signed rotation R of a bouquet G.
2: Let a := 1.
3: Step 1. If R = ∅, then stop and output the number of boundary components of G is a.
4: Step 2. Otherwise, there are three cases.
5: if there exists i such that R = AiiB then
6: Let R := AB, a := a+ 1. Return to Step 1.
7: else if there exists r such that R = IrJ(−r)P then
8: Let R := I(−J)P, a := a. Return to Step 1.
9: else
10: Find a pair of interlaced loops e, f such that R = IeJfPeQfM.
Let R := IQPJM, a := a. Return to Step 1.
Example 4.3. A bouquet with signed rotation 13214234 is not a quasi-tree bouquet. Since
there is a pair of interlaced loops 1, 3, we get a shorter signed rotation 4224. A bouquet
with signed rotation
182(−1)34325(−4)756867
is a quasi-tree bouquet. Since
182(−1)34325(−4)756867 1,(−1)−→ (−2)(−8)34325(−4)756867 (−2),2−→
(−3)(−4)(−3)85(−4)756867 (−3),(−4)−→ 85756867 8,7−→ 6565 6,5−→ ∅.
Proposition 4.4. Let G be an orientable bouquet. If G is a quasi-tree bouquet, then e(G)
is even.
Proof. It follows immediately from Euler’s formula.
Proposition 4.5. Let G be a quasi-tree bouquet. If there exists A ⊆ E(G) such that
(V (G), A) and (V (G), Ac) are plane graphs, then |A| = |Ac|.
286 Ars Math. Contemp. 22 (2022) #P2.06 / 271–286
Proof. Note that the vertex boundaries and face boundaries of Gδ(A) correspond to the face
boundaries of (V (G), A) and (V (G), Ac), respectively. It follows that v(Gδ(A)) = |A|+1
and f(Gδ(A)) = |Ac|+1, that is, Gδ(A) is a plane graph. Then E(Gδ(A)) can be partitioned
into two edge disjoint spanning trees by Lemma 3.7. Hence,
e(Gδ(A)) = 2(v(Gδ(A))− 1) = 2|A| = e(G) = |A|+ |Ac|,
that is, |A| = |Ac|.
Example 4.6. A bouquet G with signed rotation 121324356465 is not a quasi-tree bou-
quet. This follows from the fact that (G, {1, 3, 5, 6}) and (G, {2, 4}) are plane graphs, but
|{1, 3, 5, 6}| ≠ |{2, 4}|.
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.07 / 287–304
https://doi.org/10.26493/1855-3974.2443.02e
(Also available at http://amc-journal.eu)
Generalised dihedral CI-groups
Ted Dobson *
University of Primorska, UP IAM, Muzejski trg 2, SI-6000 Koper, Slovenia, and
University of Primorska, UP FAMNIT, Glagoljaşka 8, SI-6000 Koper, Slovenia
Mikhail Muzychuk
Department of Mathematics, Ben-Gurion University of the Negev, Israel
Pablo Spiga
Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca,
Via Cozzi 55, 20125 Milano, Italy
Received 24 September 2020, accepted 16 August 2021, published online 27 May 2022
Abstract
In this paper, we find a strong new restriction on the structure of CI-groups. We show
that, if R is a generalised dihedral group and if R is a CI-group, then for every odd prime
p the Sylow p-subgroup of R has order p, or 9. Consequently, any CI-group with quotient
a generalised dihedral group has the same restriction, that for every odd prime p the Sylow
p-subgroup of the group has order p, or 9.
Keywords: CI-group, DCI-group, generalised dihedral, Cayley isomorphism.
Math. Subj. Class. (2020): 05E18, 05E30
1 Introduction
Let R be a finite group and let S be a subset of R. The Cayley digraph of R with con-
nection set S, denoted Cay(R,S), is the digraph with vertex set R and with (x, y) being
an arc if and only if xy−1 ∈ S. Now, Cay(R,S) is said to be a DCI-graph (here CI
stands for Cayley isomorphic while the D stands for directed), if whenever Cay(R,S) is
isomorphic to Cay(R, T ), there exists an automorphism φ of R with Sφ = T . Clearly,
*Corresponding author. This work is supported in part by the Slovenian Research Agency (research program
P1-0285 and research projects N1-0062, J1-9108, J1-1695, N1-0140, N1-0160, J1-2451, N1-0208).
E-mail addresses: ted.dobson@upr.si (Ted Dobson), muzychuk@bgu.ac.il (Mikhail Muzychuk),
pablo.spiga@unimib.it (Pablo Spiga)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
288 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
Cay(R,S) ∼= Cay(R,Sφ) for every φ ∈ Aut(R) and hence, loosely speaking, for a DCI-
graph Cay(R,S) deciding when a Cayley digraph over R is isomorphic to Cay(R,S) is
theoretically and algorithmically elementary, but computationally efficient only if Aut(R)
is small; that is, the solving set for Cay(R,S) is reduced to simply Aut(R) (for the defini-
tion of a solving set see for example [24, 26]). The group R is a DCI-group if Cay(R,S)
is a DCI-graph for every subset S of R. Moreover, R is a CI-group if Cay(R,S) is a
DCI-graph for every inverse-closed subset S of R. Thus every DCI-group is a CI-group.
After roughly 50 years of intense research, the classification of DCI- and CI-groups is
still open. The current state of the art in this problem is as follows. There exist two rather
short lists of candidates for DCI- and CI-groups and it is known that every DCI- and every
CI-group must be a member of the corresponding list, see for instance [20]. Showing that a
candidate on the lists of possible DCI- or CI-groups is actually a DCI- or CI-group, though,
takes a considerable amount of effort. Just to give an example, the recent paper of Feng
and Kovács [15] is a tour de force that shows that elementary abelian groups of rank 5 are
DCI-groups.
In this paper we find an unexpected new restriction on which generalised dihedral
groups are CI-groups, and significantly shorten the list of candidates for CI-groups.
Definition 1.1. Let A be an abelian group. The generalised dihedral group Dih(A) over
A is the group ⟨A, x | ax = a−1,∀a ∈ A⟩. A group is called generalised dihedral if it is
isomorphic to Dih(A) for some A. When A is cyclic, Dih(A) is called a dihedral group.
Our main result is the following.
Theorem 1.2. Let Dih(A) be a generalised dihedral group over the abelian group A. If
Dih(A) is a CI-group, then, for every odd prime p the Sylow p-subgroup of A has order p,
or 9. If Dih(A) is a DCI-group, then, in addition, the Sylow 3-subgroup has order 3.
Generalised dihedral groups are amongst the most abundant members in the list of pu-
tative CI-groups. The importance of Theorem 1.2 is the arithmetical condition on the order
of such groups, which greatly reduces even further the list of candidates for CI-groups.
We believe that every generalised dihedral group satisfying this numerical condition on its
order is a genuine CI-group. (This is in line with the partial result in [8].) Additionally, this
result further reduces to two other groups on the list, whose definitions we now give.
Definition 1.3. Let A be an abelian group such that every Sylow p-subgroup of A is el-
ementary abelian. Let n ∈ {2, 4, 8} be relatively prime to |A|. Set E(A,n) = A ⋊ ⟨g⟩,
where g has order n and ag = a−1, ∀a ∈ A.
Note that E(A, 2) = Dih(A). The groups E(A, 4) and E(A, 8) have centres Z1 and
Z2 of order 2 and 4, respectively, and E(A, 4)/Z1 ∼= E(A, 8)/Z2 ∼= Dih(A). Babai
and Frankl [2, Lemma 3.5] showed that a quotient of a (D)CI-group by a characteristic
subgroup is a (D)CI-group, while the first author and Joy Morris [7, Theorem 8] showed
that a quotient of a (D)CI-group is a (D)CI-group. Applying either result and Theorem 1.2
we have the following.
Corollary 1.4. If E(A, 4) or E(A, 8) is a CI-group, then, for every odd prime p the Sylow
p-subgroup of A has order p or 9. If E(A,n), n ∈ {2, 4, 8} is a DCI-group, then, in
addition, n ̸= 8 and the Sylow 3-subgroup of A has order 3.
T. Dobson et al.: Generalised dihedral CI-groups 289
Not much is known about which of the groups under consideration in this paper are
CI-groups. Let p be a prime. Babai [1, Theorem 4.4] showed D2p is a CI-group. The first
author [4, Theorem 22] extended this to some special values of square-free integers. With
Joy Morris, the first and third authors [8] showed that D6p is a CI-group, p ≥ 5. Also, Li,
Lu, and Pálfy showed E(p, 4) and E(p, 8) are CI-groups.
We have one other result of interest, for which we will need an additional definition.
Definition 1.5. Let G be a group, and S ⊆ G. A Haar graph of G with connection set S
has vertex set G× Z2 and edge set {{(g, 0), (sg, 1)} : g ∈ G and s ∈ S}.
So a Haar graph is a bipartite analogue of a Cayley graph. There is a corresponding iso-
morphism problem for Haar graphs, and if the group A is abelian, it is equivalent to the
isomorphism problem for Cayley graphs of generalised dihedral groups Dih(A) that are
bipartite (for nonabelian groups the problems are not equivalent, as for non-abelian groups
Haar graphs need not be transitive), see [17, Lemma 2.2]. If isomorphic bipartite Cayley
graphs of Dih(A) are isomorphic by group automorphisms of A, we say A is a BCI-group.
We will also show that Zk3 is not a BCI-group for every k ≥ 3, while it is known that Zk3 is
a CI-group for every 1 ≤ k ≤ 5 [32].
1.1 Some notation
Babai [1, Lemma 3.1] has proved a very useful criterion for determining when a finite
group is a DCI-group and, more generally, when Cay(R,S) is a DCI-graph.
Lemma 1.6. Let R be a finite group, and let S be a subset of R. Then, Cay(R,S) is a
DCI-graph if and only if Aut(Cay(R,S)) contains a unique conjugacy class of regular
subgroups isomorphic to R.
Let Ω be a finite set and let G be a permutation group on Ω. An orbital graph of G is a
digraph with vertex set Ω and with arc set a G-orbit (α, β)G = {(αg, βg) | g ∈ G}, where
(α, β) ∈ Ω × Ω. In particular, each orbital graph has for its arcs one orbit on the ordered
pairs of elements of Ω, under the action of G. Moreover, we say that the orbital graphs
(α, β)G and (β, α)G are paired. When (α, β)G = (β, α)G, we say that the orbital graph is
self-paired.
When G is transitive and ω0 ∈ Ω, there exists a natural one-to-one correspondence
between the orbits of G on Ω × Ω (a.k.a. orbitals or 2-orbits of G) and the orbits of the
stabiliser Gω0 on Ω (a.k.a. suborbits of G). Therefore, under this correspondence, we may
naturally define paired and self-paired suborbits.
Two subgroups of the symmetric group Sym(Ω) are called 2-equivalent if they have the
same orbitals. A subgroup of Sym(Ω) generated by all subgroups 2-equivalent to a given
G ≤ Sym(Ω) is called the 2-closure of G, denoted G(2).
The group G is said to be 2-closed if G = G(2). It is easy to verify that G(2) is a sub-
group of Sym(Ω) containing G and, in fact, G(2) is the largest (with respect to inclusion)
subgroup of Sym(Ω) preserving every orbital of G.
290 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
2 Construction and basic results
Let q be a power of an odd prime and let F be a field of cardinality q. We let
G :=

a x z0 b y
0 0 c
 | x, y, z ∈ F, a, b, c ∈ {−1, 1}, abc = 1
 ,
D :=

a ax ax2/20 1 x
0 0 a
 | x ∈ F, a ∈ {−1, 1}
 ,
H :=

a 0 x0 a y
0 0 1
 | x, y ∈ F, a ∈ {−1, 1}
 ,
K :=

1 x y0 a 0
0 0 a
 | x, y ∈ F, a ∈ {−1, 1}
 .
It is elementary to verify that G, D, H and K are subgroups of the special linear group
SL3(F). Moreover, D, H and K are subgroups of G, |G| = 4q3, |D| = 2q and |H| =
|K| = 2q2. We summarise in Proposition 2.1 some more facts.
Proposition 2.1. The group D is generalised dihedral over the abelian group (F,+) and,
H and K are generalised dihedral over the abelian group (F⊕ F,+). The core of D in G
is 1. Moreover,
DK = DH = G = HD = KD and D ∩H = 1 = D ∩K.
Proof. The first two assertions follow with easy matrix computations. Let
g :=
1 0 00 −1 0
0 0 −1
 ∈ G
and observe that
g−1
a ax ax2/20 1 x
0 0 a
 g =
a −ax −ax2/20 1 x
0 0 a
 .
As the characteristic of F is odd, from this it follows that
D ∩Dg =
〈−1 0 00 1 0
0 0 −1
〉 .
It is now easy to see that D is core-free in G.
It is readily seen from the definitions that D ∩H = 1 = D ∩K. Therefore, |DH| =
|D||H| = 4q3 and |DK| = |D||K| = 4q3. As DH and DK are subsets of G and
|G| = 4q3, we deduce DH = G = DK and hence also HD = G = KD.
T. Dobson et al.: Generalised dihedral CI-groups 291
We let D\G := {Dg | g ∈ G} be the set of right cosets of D in G. In view of Propo-
sition 2.1, G acts faithfully by right multiplication on D\G and H and K act regularly by
right multiplication on D\G.
Proposition 2.2. The subgroups H and K are normal in G and, therefore, are in distinct
G-conjugacy classes.
Proof. The normality of H and K in G can be checked by direct computations.
2.1 Schur notation
Since G = DH and D ∩ H = 1, for every g ∈ G, there exists a unique h ∈ H with
Dg = Dh. In this way, we obtain a bijection θ : D\G → H , where θ(Dg) = h ∈ H
satisfies Dg = Dh.
Using the method of Schur (see [33]), we may identify via θ the G-set D\G with H .
Moreover, we may define an action of G on H via the following rule: for every g ∈ G and
for every h ∈ H ,
hg = h′ if and only if Dhg = Dh′.
A classic observation of Schur yields that the action of G on D\G is permutation isomor-
phic to the action of G on H . In the rest of the paper, we use both points of view.
In the action of G on H , D is a stabiliser of the identity e ∈ H , i.e. Ge = D, and H
acts on itself via its right regular representation. Since H is normal in G, the action of the
point stabiliser Ge on H is permutation equivalent to the action of Ge via conjugation on
H (Proposition 20.2 [33]). More precisely, hg = g−1hg for any g ∈ Ge and h ∈ H .
In what follows, we represent the elements of H and D as pairs [a, x] and [a, w⃗], where
x ∈ F, w⃗ ∈ F2 and a ∈ {±1}. In particular, [a, x] represents the matrixa ax ax2/20 1 x
0 0 a

of D and, if w⃗ = (x, y), then [a, w⃗] represents the matrixa 0 x0 a y
0 0 1

of H . Under this identification, the product in D and H greatly simplifies. Indeed, for
every [a, x], [b, y] ∈ D and for every [a, v⃗], [b, w⃗] ∈ H , we have
[a, x][b, y] = [ab, bx+ y], (2.1)
[a, v⃗][b, w⃗] = [ab, bv⃗ + w⃗].
Using this identification, the action of D on H also becomes slightly easier. Indeed, for
every [a, v⃗] ∈ H (with v⃗ = (x, y)) and for every [b, z] ∈ D, we have
[a, (x, y)][b,z] = [a,
(
(1− a)z2/2− byz + x, (−1 + a)z + by
)
]. (2.2)
This equality can be verified observing thata 0 x0 a y
0 0 1
b bz bz2/20 1 z
0 0 b
=
b bz bz2/20 1 z
0 0 b
a 0 (1− a)z2/2− byz + x0 a (−1 + a)z + by
0 0 1
 .
292 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
2.2 One special case
Let A := ⟨e1, e2, e3⟩, where e1 := (1 2 3), e2 := (4 5 6), e1 := (7 8 9), let
x := (1 2)(4 5)(7 8) and let R := ⟨A, x⟩. Then R is a generalised dihedral group over
the elementary abelian 3-group A of order 33 = 27. Let
S := {x, e1x, e2x, e3x, e1e2x, e21e22x, e2e3x, e22e23x, e21e22e23x}
and define
Γ := Cay(R,S).
It can be verified with the computer algebra system Magma that Aut(Γ) has order
46656 = 26 · 36, acts transitively on the arcs of Γ and (most importantly) contains two
conjugacy classes of regular subgroups isomorphic to R and hence, via Babai’s lemma, R
is not a CI-group.
This example has another interesting property from the isomorphism problem point of
view. Observe that each element of S is an involution contained in R \A. This implies that
Γ is a bipartite graph, in which case Γ is isomorphic to a Haar graph, also called a bi-coset
graph. In our example above, as every element of the connection set is an involution, it is
a Haar graph of Z33 but as it is not a CI-graph of Dih(Z33), Z33 is not a BCI-group. This is
the first example the authors are aware of where a group is an abelian DCI-group but not a
BCI-group, as Z3p is a DCI-group [3]. Our next result shows Zk3 is not a BCI-group for any
k ≥ 3.
Lemma 2.3. Let R be an abelian group and let H ≤ R. If R is BCI-group, then R/H is
BCI-group.
Proof. For this result, it is most convenient to have the vertex sets of Haar graphs and
Cayley graphs of dihedral groups be the same. So, for an abelian group R, we will have
Dih(R) permuting the set R× Z2 (the vertex set of a Haar graph of R), where an element
s ∈ R is identified with the map st : R × Z2 → R × Z2 given by st(r, i) 7→ (r + s, i).
Define ι : R × Z2 → R × Z2 by ι(r, i) = (−r, i + 1). Then Dih(R) is canonically
isomorphic to G = ⟨ι, st : s ∈ R⟩. It is straightforward to show that ι ∈ Aut(Haar(R,S)),
and so we have G ≤ Aut(Haar(R,S)) for every S ⊆ R. By [28, Theorem 2], we have
Haar(R,S) ∼= Cay(Dih(R), T ), for some T ⊆ G, by the map ϕ which identifies (r, i) with
the unique element of G which maps (0, 0) to (r, i), r ∈ R, i ∈ Z2. Hence ϕ(r, i) = rtιi,
and T = {sι : s ∈ S} = S · ι.
If R is a BCI-group, then Haar(R,S) is a BCI graph. Let C = {R×{0}, R×{1}}, B
be the set of right cosets of H in Dih(R), and U = {sH : s ∈ S}. Then, as partitions of
R × Z2, B refines C. As C is a bipartition of Cay(Dih(R), S · ι), Cay(Dih(R/H), U · ι)
is bipartite with bipartition {{(rH, i) : r ∈ R} : i ∈ Z2} and so Cay(Dih(R/H), U · ι) =
Haar(R/H,U).
As Cay(Dih(R), S · ι) is a CI-graph of Dih(R), by the proof of [6, Theorem 8],
we see Cay(Dih(R/H), U · ι) is a CI-graph of Dih(R/H) and any Cayley graph of
Dih(R/H) isomorphic to Cay(Dih(R/H), U · ι) is isomorphic by a group automorphism
of Dih(R/H). But this means any two Haar graphs of R/H are isomorphic by a group
automorphism of Dih(R/H), and so R/H is a BCI-group.
Finally, Γ, as well as the graphs constructed in the next section, have the property
that the Sylow p-subgroups of their automorphism groups are not isomorphic to Sylow p-
subgroups of any 2-closed group of degree 33 or p2 (in the next section). For the example
T. Dobson et al.: Generalised dihedral CI-groups 293
above, the Sylow p-subgroups of the automorphism groups of Cayley digraphs of Z3p can
be obtained from [5, Theorem 1.1], and none have order 36 as a Sylow p-subgroup of
AGL(3, 3) is not 2-closed (for p2 in the next section, the Sylow p-subgroup has order p3,
but Sylow p-subgroups of the automorphism groups of Cayley digraphs of Z2p have order
p2 or pp+1 [10, Theorem 14]).
3 The permutation group G is 2-closed
In this section we prove the following.
Proposition 3.1. The group G in its action on H is 2-closed.
We start with some preliminary observations.
Lemma 3.2. The orbits of Ge on H have one of the following forms:
(1) St := {[1, (t, 0)]}, for every t ∈ F;
(2) Ct ∪ C−t, where Ct := {[1, (z, t)] | z ∈ F} and t ∈ F \ {0};
(3) Pt :=
{
[−1, (t+ z2, 2z)] | z ∈ F
}
with t ∈ F.
Proof. Let g := [a, (x, y)] ∈ H . If a = 1 and y = 0, then (2.2) yields
g[b,z] = [1, (x, 0)] = g
and hence the Ge-orbit containing g is simply {g}. Therefore we obtain the orbits in
Case (1).
Suppose then a = 1 and y ̸= 0. Now, 2.2 yields
g[1,z] = [1, (−yz + x, y)],
g[−1,z] = [1, (yz + x,−y)].
In particular, Cy = {g[1,z] | z ∈ F} and C−y = {g[−1,z] | z ∈ F} and we obtain the orbits
in Case (2).
Finally suppose a = −1. Now, (2.2) yields
g[b,z] = [1, (z2 − byz + x,−2z + by)].
In particular, if we choose z := by/2 and t = −y2/4 + x, then g and [−1, (t, 0)] are in the
same Ge-orbit. Therefore [−1, (x, y)]Ge = [−1, (t, 0)]Ge . Using again (2.2), we get
[−1, (t, 0)][b,−z] = [−1, (t+ z2, 2z)].
In particular, Pt = {g[b,z] | [b, z] ∈ Ge} and we obtain the orbits in Case (3).
We call the Ge-orbits in (1) singleton orbits, the Ge-orbits in (2) coset orbits and the
Ge-orbits in (3) parabolic orbits. Clearly, singleton orbits have cardinality 1, coset orbits
have cardinality 2q and parabolic orbits have cardinality q. Also, it follows from Lemma 3.2
that there are q singleton orbits, q−12 coset orbits and q parabolic orbits. Indeed,
q · 1 + q − 1
2
· 2q + q · q = 2q2 = |H|.
It is also clear from Lemma 3.2 that all non-singleton orbits are self-paired and the only
self-paired singleton orbit is S0.
Before continuing, we recall [14, Definitions 2.5.3 and 2.5.4] tailored to our needs.
294 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
Definition 3.3. We say that h ∈ H separates the pair (h1, h2) ∈ H × H , if (h, h1) and
(h, h2) belong to distinct G-orbitals, that is, hh−11 and hh
−1
2 are in distinct Ge-orbits.
We also say that a subset S ⊆ H separates G-orbitals if, for any two distinct elements
h1, h2 ∈ H \ S, there exists s ∈ S separating the pair (h1, h2).
Proposition 3.4. If q ≥ 5, then {e} ∪ P0 separates G-orbitals.
Proof. Set S := {e} ∪ P0. Let h1, h2 ∈ H \ S be two distinct elements. If h1 and h2
belong to distinct Ge-orbits, then e ∈ S separates (h1, h2). Therefore, we assume that h1
and h2 belong to the same Ge-orbit, say, O. Since h1 ̸= h2, O is not a singleton orbit and
hence O is either a coset or a parabolic orbit.
Assume first that O is a parabolic orbit, that is, O = Pt, for some t ∈ F. By Lemma 3.2,
for each i ∈ {1, 2}, there exists xi ∈ F with hi = [−1, (t + x2i , 2xi)]. As q = |F| ≥ 5, it
is easy to verify that there exists x ∈ F with x /∈ {x1, x2} and with x− x1 ̸= −(x− x2).
Now, let s := [−1, (x2, 2x)] ∈ P0 ⊆ S. From (2.1), we deduce
sh−1i = [1, (t+ x
2
i − x2, 2xi − 2x)].
As 2xi−2x ̸= 0, from Lemma 3.2, we obtain sh−1i ∈ C2(x−xi)∪C−2(x−xi). As x−x1 ̸=
−(x− x2), we deduce that sh−11 and sh
−1
2 are in distinct Ge-orbits and hence s separates
(h1, h2).
Assume now that O is a coset orbit, that is, O = Ct ∪ C−t, for some t ∈ F \ {0}. In
this case, for each i ∈ {1, 2}, there exist xi ∈ F and ai ∈ {±1} with hi = [1, (xi, ait)].
Let x ∈ F with
xt(a2 − a1) ̸= x2 − x1.
(The existence of x is clear when a1 ̸= a2 and it follows from the fact that h1 ̸= h2 when
a1 = a2.) Set s := [−1, (x2, 2x)] ∈ P0 ⊆ S. From (2.1), we have
sh−1i ∈ [−1, (x
2 − xi, 2x− ait)].
In particular, from Lemma 3.2, we have sh−1i ∈ Pti , for some ti ∈ F. Thus, (x2−xi, 2x−
ait) = (ti + y
2, 2y), for some y ∈ F. From this it follows that
ti = x
2 − xi −
(2x− ait)2
4
.
As xt(a2 − a1) ̸= x2 −x1, a simple computation yields t1 ̸= t2 and hence sh−11 and sh
−1
2
are in distinct Ge-orbits. Therefore, s separates (h1, h2).
Proof of Proposition 3.1. When q = 3, the proof follows with a computation with the
computer algebra system Magma. Therefore, for the rest of the proof we suppose q ≥
5. Let T be the 2-closure of G. As {e} ∪ P0 separates the G-orbitals, it follows from
[14, Theorem 2.5.7] that the action of Te on P0 is faithful, and hence so is the action of
Ge on P0. We denote by GP0e (respectively, T
P0
e ) the permutation group induced by Ge
(respectively, Te) on P0. In particular, Ge ∼= GP0e and Te ∼= TP0e .
We claim that
(Te)
P0 = (Ge)
P0 . (3.1)
Observe that from (3.1) the proof of Proposition 3.1 immediately follows. Indeed, Te ∼=
TP0e = G
P0
e
∼= Ge and hence Te = Ge. As H is a transitive subgroup of G, we deduce that
T. Dobson et al.: Generalised dihedral CI-groups 295
G = GeH = TeH = T and hence G is 2-closed. Therefore, to complete the proof, we
need only establish (3.1).
From Lemma 3.2, |P0| = q. Hence (Ge)P0 is a dihedral group of order 2q in its natural
action on q points.
For each t ∈ F∗ let Φt be the subgraph of Cay(H,Ct ∪ C−t) induced by P0. Let
(h1, h2) be an arc of Φt. As h1, h2 ∈ P0, there exist x1, x2 ∈ F with h1 = [−1, (x21, 2x1)]
and h2 = [−1, (x22, 2x2)]. Moreover, h2h−11 ∈ Ct ∪ C−t and hence, by (2.1), we obtain
h2h
−1
1 = [1, (x
2
2 − x21, 2x2 − 2x1)] ∈ Ct ∪ C−t,
that is, 2x2 − 2x1 ∈ {−t, t}. This shows that the mapping
P0 → F+
(x2, 2x) 7→ 2x
is an isomorphism between the graphs Φt and Cay(F+, {−t, t}). Therefore
(Ge)
P0 ≤ (Te)P0 ≤
⋂
t∈F∗
Aut(Φt) ∼=
⋂
t∈F∗
Aut(Cay(F+, {−t, t})) ∼= Dih(F+).
Since (Ge)P0 and Dih(F+) are dihedral groups of order 2q, we conclude that (Ge)P0 =
(Te)
P0 =
⋂
t∈F∗ Aut(Φt), proving 3.1.
4 Generating graph
Combining Proposition 3.1, Proposition 2.2, and Lemma 1.6, we have proven that Dih(Z2p)
is not a CI-group with respect to colour Cayley digraphs for odd primes p. In this section
we strengthen that result to Cayley graphs.
4.1 Schur rings
Let R be a finite group with identity element e. We denote the group algebra of R over the
field Q by QR. For Y ⊆ R, we define
Y :=
∑
y∈Y
y ∈ QR.
Elements of QR of this form will be called simple quantities, see [33]. A subalgebra A of
the group algebra QR is called a Schur ring over R if the following conditions are satisfied:
(1) there exists a basis of A as a Q-vector space consisting of simple quantities
T 0, . . . , T r;
(2) T0 = {e}, R =
⋃r
i=0 Ti and, for every i, j ∈ {0, . . . , r} with i ̸= j, Ti ∩ Tj = ∅;
(3) for each i ∈ {0, . . . , r}, there exists i′ such that Ti′ = {t−1 | t ∈ Ti}.
Now, T 0, . . . , T r are called the basic quantities of A. A subset S of R is said to be an
A- subset if S ∈ A, which is equivalent to S =
⋃
j∈J Tj , for some J ⊆ {0, . . . , r}.
Given two elements a :=
∑
x∈R axx and b :=
∑
y∈R byy in QR, the Schur-Hadamard
product a ◦ b is defined by
a ◦ b :=
∑
z∈R
azbzz.
296 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
It is an elementary exercise to observe that, if A is a Schur ring over R, then A is closed by
the Schur-Hadamard product.
The following statement is known as the Schur-Wielandt principle, see [33, Proposi-
tion 22.1].
Proposition 4.1. Let A be a Schur ring over R, let q ∈ Q and let x :=
∑
r∈R arr ∈ A.
Then
xq :=
∑
r∈R
ar=q
r ∈ A.
Let X be a permutation group containing a regular subgroup R. As in Section 2.1, we
may identify the domain of X with R. Let T0, . . . , Tr be the orbits of Xe with T0 = {e}.
A fundamental result of Schur [33, Theorem 24.1] shows that the Q-vector space spanned
by T 0, T 1, . . . , T r in QR is a Schur ring over R, which is called the transitivity module of
the permutation group X and is usually denoted by V (R,Ge). In particular, the V (R,Ge)-
subsets of the Schur ring V (R,Ge) are unions of Ge-orbits.
Let A := ⟨T 0, . . . , T r⟩ be a Schur ring over R (where T0, . . . , Tr are the basic quanti-
ties spanning A). The automorphism group of A is defined by
Aut(A) :=
r⋂
i=0
Aut(Cay(R, Ti)). (4.1)
Given a subset S of R, we denote by
⟨⟨S⟩⟩,
the smallest (with respect to inclusion) Schur ring containing S. Now, ⟨⟨S⟩⟩ is called the
Schur ring generated by S.
We conclude this brief introduction to Schur rings recalling [25, Theorem 2.4].
Proposition 4.2. Let S be a subset of R. Then Aut(⟨⟨S⟩⟩) = Aut(Cay(R,S)).
4.2 The group G is the automorphism group of a single (di)graph
It was shown above that the group G is 2-closed, i.e. it is the automorphism of a coloured di-
graph. In this section we give a Cayley digraph Cay(H,T ) having automorphism group G.
To build such a digraph it is sufficient to find a subset T ⊆ H such that
⟨⟨T ⟩⟩ = V (H,Ge) (Proposition 4.2). Such a set is constructed in Proposition 4.3. Note
that T is symmetric for q ≥ 7, so the digraph Cay(H,T ) is undirected. The cases of
q = 3, 5 are exceptional, because in those cases no inverse-closed subset of H has the
required property.
Proposition 4.3. Let q be prime, and
T :=

P0 ∪ P1 ∪ Px ∪ C1 ∪ C−1 where x ∈ F with x ̸∈ {0,±1,±2, 12} and x
6 ̸= 1,
when q > 7,
P0 ∪ P1 ∪ P3 ∪ C1 ∪ C−1 when q = 7,
S1 ∪ P0 when q = 5,
S1 ∪ P0 when q = 3.
Then ⟨⟨T ⟩⟩ = V (H,Ge). In particular, T is not a (D)CI-subset of H .
T. Dobson et al.: Generalised dihedral CI-groups 297
Proof. When q ≤ 7, the result follows by computations with the computer algebra system
Magma. Therefore for the rest of the proof we suppose q > 7.
According to Proposition 3.2 the basic sets of V (H,Ge) are of three types: Sa, Cb ∪
C−b, Pc with a, b, c ∈ F and b ̸= 0. Thus we have three types of basic quantities Sa,
Cb + C−b, Pc and
V (H,Ge) = ⟨Sa, Cb + C−b, Pc a, b, c ∈ F, b ̸= 0⟩.
Set
H1 := {[1, v⃗] | v⃗ ∈ F2},
H2 := {[1, (t, 0)] | t ∈ F}.
By (2.1), H1 and H2 are subgroups of H with |H2| = q, |H1| = q2 and, by Lemma 3.2,
H2 = ∪t∈FSt. In Table 4.2 we have reported the multiplication table among the basic
quantities of V (H,Ge): this will serve us well.
Sr Cs Pt
Sa Sa+r Cs Pt−a
Cb Cb
{
qCb+s if b+ s ̸= 0
qH2 if b+ s = 0
H \H1
Pc Pc+r H \H1 qS−c+t +H1 \H2
Table 1: Multiplication table for the basic quantities of V (H,Ge).
Fix a, b, c ∈ F with b, c ̸= 0 and let A be the smallest Schur ring of the group algebra
QH containing Pa, Cb + C−b, Sc. We claim that
A = V (H,Ge). (4.2)
Clearly, A ≤ V (H,Ge). From Table 4.2, for every k ∈ {0, . . . , q−1}, we have Sck = Sck
and hence Sck ∈ A. As c ̸= 0, Si ∈ A, for each i ∈ {0, . . . , q − 1}. Now, as Pa ∈ A,
from Table 4.2, we have Pa · Si = Pa+i ∈ A for any i ∈ {0, . . . , q − 1}. The equality
(Cb+C−b)
2 = 2qH2+ qC2b+ qC−2b implies C2b+C−2b ∈ A. Now arguing inductively
we deduce Ck + C−k ∈ A, for all k ∈ {1, . . . , q − 1}. Thus (4.2) follows.
Let x ∈ F with x ̸∈ {0,±1,±2, 12} and x
6 ̸= 1, let T := P0 ∪ P1 ∪ Px ∪ C1 ∪ C−1
and let T := ⟨⟨T ⟩⟩ (the existence of x is guaranteed by the fact that q > 7). We claim that
H2, H1, C2 + C−2, S1 + S−1 + Sx + S−x + S1−x + Sx−1 ∈ T . (4.3)
Using Table 4.2 for squaring T , we obtain (after rearranging the terms):
T 2 =3qS0 + qS1 + qS−1 + qSx + qS−x + qS1−x + qSx−1
+ 9H1 \H2 + 12H \H1 + qC2 + qC−2 + 2qH2.
298 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
From the assumptions on x, the elements −1, 1,−x, x,−(x−1), x−1 are pairwise distinct.
Therefore
T 2 ◦ Sb =

5qS0, b = 0,
3qSb, if b ∈ {±1,±x,±(x− 1)},
2qSb, if b ̸∈ {0,±1,±x,±(x− 1)},
T 2 ◦ Cb =
{
(q + 9)Cb, if b ∈ {±2},
9Cb, if b ̸∈ {0,±2},
T 2 ◦ Pb = 12Pb, if b ∈ F.
Since the numbers 6, 9, q + 9, 2q, 3q, 5q are also pairwise distinct (because q ̸= 3), an
application of the Schur-Wielandt principle yields
(T 2)3q = S1 + S−1 + Sx + S−x + S1−x + Sx−1 ∈ T ,
(T 2)12 = H \H1 ∈ T ,
(T 2)2q = H2 − (S0 + S1 + S−1 + Sx + S−x + S1−x + Sx−1) ∈ T ,
(T 2)q+9 = C2 + C−2 ∈ T .
From this, (4.3) immediately follows.
We claim that
S1 + S−1 ∈ T . (4.4)
Let
TH2 := T ∩QH2
and observe that TH2 is a Schur ring over the cyclic group H2 ∼= Zq of prime order q. It is
well known that every Schur ring over Zq is determined by a subgroup M ≤ Aut(Zq) ∼=
Z∗q such that, every basic set of the corresponding Schur ring is an M -orbit. Let M be such
a subgroup for TH2 . From (4.3), the simple quantity S1 + S−1 + Sx + S−x + S1−x +
Sx−1 belongs to TH2 and hence {±1,±x,±(1 − x)} is a TH2 -subset of cardinality 6.
It follows that |M | divides six and M ⊆ {±1,±x,±(1 − x)}. If |M | ∈ {3, 6}, then
{±1,±x,±(1− x)} is a subgroup of Z∗q , contrary to the assumption x6 ̸= 1. Therefore
either M = {1} or |M | = {±1}. (4.5)
In both cases, {−1, 1} is a union of M -orbits. Therefore, S1+S−1 ∈ TH2 . From this, (4.4)
follows immediately.
We are now ready to conclude the proof. Clearly, T ∈ V (H,Ge) and hence T ⊆
V (H,Ge). From (4.3), H1 ∈ T and, from (4.4), S1 + S−1 ∈ T . Therefore H1 ◦ T =
C1 + C−1 ∈ T and (T −H1) ◦ T = P0 + P1 + Px ∈ T . Therefore(
(P0 + P1 + Px)(S1 + S−1)
)
◦ (P0 + P1 + Px) ∈ T .
As (P0 + P1 + Px)(S1 + S−1) = P1 + P2 + Px+1 + P−1 + P0 + Px−1, we deduce(
(P0 + P1 + Px)(S1 + S−1)
)
◦ (P0 + P1 + Px) = P0 + P1
T. Dobson et al.: Generalised dihedral CI-groups 299
and hence P0 + P1 ∈ T . Therefore, Px = (P0 + P1 + Px)− (P0 + P1) ∈ T . As
(P0 + P1)Px = qSx + qSx−1 + 2(H \H1),
from the Schur-Wielandt principle, we obtain Sx+Sx−1 ∈ T . Therefore Sx+Sx−1 ∈ TH2
and hence {x, x − 1} is a TH2 -subset. Thus {x, x − 1} is an M -orbit. Recall (4.5). If
M = {−1, 1}, then x− 1 = −1 · x = −x, contrary to the assumption x ̸= 1/2. Therefore
M = {1} and TH2 = QH2. Thus Si ∈ T , for each i ∈ Zq . Thus S1, Px, C1 + C−1 ∈ T
and (4.2) implies V (H,Ge) ⊆ T .
5 Proof of Theorem 1.2
Proof of Theorem 1.2. The list of candidate CI-groups is on page 323 in [20]. From here,
we see that, if R is in this list and if R = Dih(A) is generalised dihedral, then for every
odd prime p the Sylow p-subgroup of R is either elementary abelian or cyclic of order 9.
Assume that the Sylow p-subgroup (p is an odd prime) of A is elementary abelian of
rank at least 2. Let P ≤ A be a subgroup isomorphic to Z2p and let x ∈ R\A. Then ⟨P, x⟩ ∼=
Dih(Z2p). By Proposition 4.3, Dih(Z2p) contains a non-DCI subset. Therefore Dih(Z2p) is
a non-DCI-group. Since subgroups of a (D)CI-group are also (D)CI, we conclude that R is
a not a DCI-group as well. The non-DCI set T constructed in Proposition 4.3 is symmetric
for p ≥ 7. Hence Dih(Z2p) and, therefore, R are non-CI groups when p ≥ 7. If p = 5, then
the group Dih(Z2p) contains a non-CI subset, namely: P0 ∪ S1 ∪ S−1 (this was checked by
Magma1). Combining these arguments we conclude that if Dih(A) is a CI-group, then its
Sylow p-subgroup is cyclic if p ≥ 5. If p = 3, then the Sylow 3-subgroup is either cyclic
of order 9 or elementary abelian. The example in Section 2.2 shows that the rank of an
elementary abelian group is bounded by 2.
We now give the updated list of CI-groups. It is a combination of the list in [20],
together with our results here and [12, Corollary 13] (note [12, Corollary 13] contains an
error, and should list Q8 on line (1c), not on line (1b)). We need to define one more group:
Definition 5.1. Let M be a group of order relatively prime to 3, and exp(M) be the largest
order of any element of M . Set E(M, 3) = M ⋊ϕZ3, where ϕ(g) = gℓ, and ℓ is an integer
satisfying ℓ3 ≡ 1 (mod exp(M)) and gcd(ℓ(ℓ− 1), exp(M)) = 1.
Theorem 5.2. Let G, M , and K be CI-groups with respect to graphs such that M and K
are abelian, all Sylow subgroups of M are elementary abelian, and all Sylow subgroups of
K are elementary abelian of order 9 or cyclic of prime order.
(1) If G does not contain elements of order 8 or 9, then G = H1 ×H2 ×H3, where the
orders of H1, H2, and H3 are pairwise relatively prime, and
(a) H1 is an abelian group, and each Sylow p-subgroup of H1 is isomorphic to Zkp
for k < 2p+ 3 or Z4;
(b) H2 is isomorphic to one of the groups E(K, 2), E(M, 3), E(K, 4), A4, or 1;
(c) H3 is isomorphic to one of the groups D10, Q8, or 1.
1The automorphism group of the corresponding Cayley graph is 4 times bigger than G but the subgroups H
and K are non-conjugate inside it.
300 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
(2) If G contains elements of order 8, then G ∼= E(K, 8) or Z8.
(3) If G contains elements of order 9, then G is one of the groups Z9 ⋊ Z2, Z9 ⋊ Z4,
Z22 ⋊ Z9, or Zn2 × Z9, with n ≤ 5.
Remark 5.3. The rank bound of an elementary abelian group used in part (1)(a) is due to
[29].
Other than positive results already mentioned, the abelian groups known to be CI-
groups are Z2n [22], Z4n [23] with n an odd square-free integer, Zq × Z2p [18],
Zq × Z3p [31], and Zq × Z4p [19] with q and p and distinct primes, and Z32 × Zp [9]. Addi-
tional results are given in [4, Theorem 16] and [11] with technical restrictions on the orders
of the groups. A similar result with technical restrictions on M is given in [4, Theorem
22] for some E(M, 3). Also, E(Zp, 4) and E(Zp, 8) were shown to be CI-groups in [21],
and Q8 × Zp in [30]. Finally, Holt and Royle have determined all CI-groups of order at
most 47 [16]. Applying Theorem 5.2 to determine possible CI-groups, and then checking
the positive results above to see that all possible CI-groups are known to be CI-groups, we
extend the census of CI-groups up to groups of order at most 59. The isomorphism problem
for circulant digraphs was independently solved in [13] and [26] (in both cases a polyno-
mial time algorithm for solving the isomorphism problem was given). A polynomial time
algorithm for finding the automorphism group of circulant digraph was provided in [27].
Finally, we remark that the groups E(M, 3) and E(M, 8) are not DCI-groups.
Appendix A An alternative approach
In this section we give an alternative approach to the proof of Theorem 1.2. We do not
give all of the details - just the basic idea. In principle, this section is independent from the
previous sections, but for convenience we deduce the main result from our previous work.
For each g ∈ GL3(F), let g⊤ denote the transpose of the matrix g and let gι := (g−1)⊤.
It is easy to verify that ι : GL3(F) → GL3(F) is an automorphism. Let
s =
0 0 10 1 0
1 0 0

and let α be the automorphism of GL3(F) defined by
gα := s−1gιs = s−1(g−1)⊤s, (A.1)
for every g ∈ GL3(F).
We now define α̂ ∈ Sym(H) by
[a, (x, y)]α̂ = [a, (y2/2− x, ay)], (A.2)
for every [a, (x, y)] ∈ H .
Lemma A.1. Let α and α̂ be as in (A.1) and (A.2). We have
(1) Gα = G and Dα = D;
(2) K = Hα and H = Kα;
T. Dobson et al.: Generalised dihedral CI-groups 301
(3) for every h ∈ H , (Dh)α = Dhα̂;
(4) for every x ∈ F and for every t ∈ F∗, Sα̂x = S−x, Cα̂t = Ct, P α̂x = P−x.
Proof. The proof follows from straightforward computations. For every a ∈ {−1, 1} and
x ∈ F, we havea ax ax2/20 1 x
0 0 a
α =
0 0 10 1 0
1 0 0


a ax ax2/20 1 x
0 0 a
−1

⊤ 0 0 10 1 0
1 0 0

=
0 0 10 1 0
1 0 0
a −x a(−x)2/20 1 a(−x)
0 0 a
⊤ 0 0 10 1 0
1 0 0

=
0 0 10 1 0
1 0 0
 a 0 0−x 1 0
a(−x)2/2 a(−x) a
0 0 10 1 0
1 0 0

=
a a(−x) a(−x)2/20 1 −x
0 0 a
 ∈ D.
This shows Dα = D. The computations for proving G = Gα, K = Hα and H = Kα
are similar.
Let h := [a, (x, y)] ∈ H . A direct computation shows that
hα =
a 0 x0 a y
0 0 1
α =
1 −ay −ax0 a 0
0 0 a

and hence
hα(hα̂)−1 =
1 −ay −ax0 a 0
0 0 a
a 0 y2/2− x0 a ay
0 0 1
−1
=
1 −ay −ax0 a 0
0 0 a
a 0 −ay2/2 + ax0 a −y
0 0 1

=
a −y ay2/20 1 −ay
0 0 a
 ∈ D.
Therefore
(Dh)α = Dαhα = Dhα = Dhα̂
and part (3) follows. Now, part (4) follows immediately from Lemma 3.2 and part (3).
Lemma A.2. Let x ∈ F with x ̸∈ {0,±1,±2, 12} and x
6 ̸= 1, and let
T := P0 ∪ P1 ∪ Px ∪ C1 ∪ C−1,
T ′ := P0 ∪ P−1 ∪ P−x ∪ C1 ∪ C−1.
302 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
Then Cay(H,T ) and Cay(H,T ′) are isomorphic but not Cayley isomorphic. In particular,
H is not a CI-group.
Proof. We view G as a permutation group on D\G, which we may identify with H via the
Schur notation.
It follows from Lemma A.1(1) and (3) that α̂ normalizes G. Therefore, α̂ permutes
the orbitals of G. Since α̂ fixes e = [1, (0, 0)], α̂ permutes the suborbits of G and, from
Lemma A.1(4), we have Cay(H,T α̂) = Cay(H,T ′). Hence Cay(H,T )α̂ = Cay(H,T ′)
and Cay(H,T ) ∼= Cay(H,T ′).
Assume that there exists β ∈ Aut(H) with Cay(H,T )β = Cay(H,T ′). Then α̂β−1
is an automorphism of Cay(H,T ). It follows from Propositions 4.2 and 4.3 that α̂β−1 ∈
Aut(Cay(H,T )) = G. Therefore α̂ ∈ Gβ. Since G and β normalize H , so does α.
However, this contradicts Lemma A.1(2).
On the previous proof, one could prove directly that there exists no automorphism β of
H with T β = T ′; however, this requires some detailed computations, in the same spirit as
the computations in Section 4.2.
ORCID iDs
Mikhail Muzychuk https://orcid.org/0000-0002-6346-8976
Pablo Spiga https://orcid.org/0000-0002-0157-7405
Ted Dobson https://orcid.org/0000-0003-2013-4594
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.08 / 305–315
https://doi.org/10.26493/1855-3974.2584.68d
(Also available at http://amc-journal.eu)
The antiprism of an abstract polytope*
Ian Gleason
Univeristy of California, Berkeley, United States
Isabel Hubard †
Instituto de Matemáticas, Universidad Nacional Autonoma de México,
Circuito Exterior, C.U. Coyoacán 04510, México D.F.
Received 21 March 2021, accepted 19 August 2021, published online 14 June 2022
Abstract
Antiprisms of polygons are classical convex vertex-transitive polyhedra. In this paper,
for any given (abstract) polytope, we define its antiprism. We then find the automorphism
group of the antiprism of P in terms of the extended group of P (the groups of auto-
morphisms and dualities) as well as some transitivity properties. We also give a relation
between some products of abstract polytopes and their antiprisms.
Keywords: Antiprism, abstact polytopes.
Math. Subj. Class. (2020): 51M20, 52B05
The antiprism is a classical convex polyhedron. The antiprism of a polygon can be
constructed by taking, in Euclidian 3-space, two identical copies of a regular n-gon in
parallel planes, in such a way that the vertices of one of the polygons are “aligned” with
the mid points of the edges of the other. By taking the convex hull of all the vertices, we
obtain the antiprism over an n-gon (see Figure 1).
For higher dimensions, the concept of a convex antiprism is not always defined (see [1,
2] and [3] for further discussion of the subject). In this paper we define the antiprism of any
abstract polytope and show that it is indeed again an abstract polytope. The given definition
generalizes the antiprism of a polygon and satisfies that the antiprism of a polytope and its
dual is the same.
The paper uses some of the ideas and notation of the products of polytopes described
in [4]. Moreover, we give relations between some products and their antiprisms. We then
use such relations to compute the automorphism group of an antiprism. These results are
summarized in the following theorem.
*The authors would like to thank Steve Wilson for suggesting studying the antiprism of an abstract polytope.
We also gratefully acknowledge financial support of the PAPIIT-DGAPA, under grant IN-109218 and of CONA-
CyT, under grant A1-S-21678.
†Corresponding author.
E-mail addresses: ianandreigf@gmail.com (Ian Gleason), isahubard@im.unam.mx (Isabel Hubard)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
306 Ars Math. Contemp. 22 (2022) #P2.08 / 305–315
Figure 1: Antiprism over a pentagon.
Theorem A. Let P and Q be two abstract polytopes and Ant(P),Ant(Q) be their an-
tiprisms.
(a) If ⋊⋉ and ⊕ denote the join product and the direct sum of abstract polytopes, respec-
tively, then Ant(P ⋊⋉ Q) ∼= Ant(P)⊕Ant(Q).
(b) If Γ̂ denotes the extended group of P and P = Qm11 ⋊⋉ Q
m2
2 ⋊⋉ · · · ⋊⋉ Qmrr , where
each Qi is a prime polytope with respect to the join product, then
Γ(Ant(P)) = Πri=1
(
(Γ̂(Qi))mi ⋊ Smi
)
.
In particular, if P is prime with respect to the join product, then Γ(Ant(P)) ∼= Γ(P)
whenever P is a not a self-dual polytope, while if P is self-dual, then Γ(P) has index
2 in Γ(Ant(P)) = Γ̂(P).
(c) Let P be a prime polytope with respect to the join product. If P is a k-orbit polytope
of rank n, then Ant(P) is either a 2nk-orbit polytope (if P is self-dual) or a 2n+1k-
orbit polytope (if P is not self-dual).
The paper is organized as follows. Section 1 reviews the basic notions about abstract
polytopes and their products. In Section 2 we define the antiprism and show that is always
an abstract polytope and analyse the flags of the antiprism in terms of the flags of the poly-
tope. Sections 3 and 4 deal with the interaction between some products and the antiprism,
and with the study of the automorphism group of an antiprism, respectively.
1 Abstract polytopes, their join product and direct sum
Abstract polytopes are combinatorial generalizations of the face lattice of convex poly-
topes. In this section we give the basic definitions from the theory of abstract polytopes, as
well as two of their products. For details on these subjects we refer the reader to [5] and
[4], respectively.
An (abstract) polytope is a partially ordered set (poset) P , whose elements are called
faces, such that it has a minimal and a maximal element and is ranked: all its maximal
chains, called flags, have the same number of elements. This endows the poset with a rank
function r satisfying that if F,G ∈ P with F ≤ G, then r(F ) ≤ r(G), and if r(F ) = r(G),
then F and G are either equal or they are not incident in P . We say that the minimal face
has rank −1, and if the range of the rank function is {−1, 0, . . . , n}, then we say that P
has rank n or is a n-polytope. A face of rank i is said to be an i-face and the 0-, 1- and
n− 1-faces are the vertices, edges and facets of P , respectively. The minimal and maximal
faces are the improper faces of P , and all other faces are proper. We also require that P
satisfies the diamond condition, meaning that whenever F,G ∈ P are two incident faces
I. Gleason and I. Hubard: The antiprism of an abstract polytope 307
such that their ranks differ by 2, then there are exactly two faces H,H ′ of P satisfying that
F < H,H ′ < G. Finally, we ask that P be strongly connected in the sense that the poset is
connected and each of its open intervals with more than two elements is connected as well.
A section of P is a closed interval of P . Every section of P is a polytope in its own
right. The diamond condition is equivalent to saying that all sections of rank 1 have exactly
4 faces. This condition also implies that for each i ∈ {0, 1, . . . , n − 1} and every flag Φ,
there is a unique i-adjacent flag to Φ that differs from Φ only in the element of rank i. We
shall denote the set of all flags of P by F(P), and the i-adjacent flag of Φ by Φi.
The dual of a polytope P is the poset that has the same elements as P , but with the
reverse order. If a polytope is isomorphic to its dual, it is said to be self-dual.
An automorphism of P is an order preserving bijection. The group of all automor-
phisms of P is its automorphism group and it shall be denoted by Γ(P). A duality of a
self-dual polytope is an order reversing bijection. The composition of two dualities of a
self-dual polytope is not a duality, but an automorphim. Thus, the extended group of P
is the group that contains all automorphisms and dualities of P and it will be denoted by
Γ̂(P). Note that Γ̂(P) has Γ(P) as a subgroup of index at most 2; the groups coincide
whenever P is not self-dual.
Given two polytopes P and Q, their join product, P ⋊⋉ Q, is the polytope whose
elements are the pairs (F,G), with F ∈ P and G ∈ Q. Two elements (F,G) and (F ′, G′)
are incident in P ⋊⋉ Q if and only if F ≤P F ′ and G ≤Q G′. The rank of (F,G) is
rankP(F ) + rankQ(G) + 1. A polytope P is said to be prime with respect to the join
product if it cannot be decomposed as the join product of two polytopes of ranks at least 0.
The direct sum of the polytopes P and Q, with maximum elements Fn andGm, respec-
tively is P⊕Q = {(F,G) ∈ P ⋊⋉ Q | F ̸= Fn, G ̸= Gm}∪{(Fn, Gm)}. The order of the
direct sum is given by (F,G) ≤P⊕Q (F ′, G′) if and only if F ≤P F ′ and G ≤Q G′, and
the rank of the face (F,G) is rankP(F )+rankQ(G), which implies that the rank of P⊕Q
is n +m. A polytope P is said to be prime with respect to the direct sum if it cannot be
decomposed as the join product of two polytopes of ranks at least 1. The following lemma
falls straightforward from the definitions.
Lemma 1.1. Let P and Q be two polytopes, and let (F,G) be a proper face of P ⊕ Q.
Then the section {(H,K) ∈ P ⊕ Q | (H,K) ≤P⊕Q (F,G)} is isomorphic to the join
product of the sections {H ∈ P | H ≤P F} and {K ∈ Q | K ≤Q G}.
In [4] the authors study the automorphism group of a product in terms of the automor-
phisms groups of the factors. In particular we have the following result.
Theorem 1.2 ([4]). Let P = Qm11 ⊕Q
m2
2 ⊕· · ·⊕Qmrr , where each Qi is a prime polytope
with respect to the direct sum. Then
Γ(P) = Πri=1
(
(Γ(Qi))mi ⋊ Smi
)
.
2 The antiprism
The antiprism of a polygon is a convex polyhedron in ordinary 3-space. Its faces are two
regular n-gons and 2n equilateral triangles. When n = 3, we obtain the regular octahedron.
Otherwise, the antiprism over an n-gon is an Archemidian solid, as their faces are all regular
polygons and its group of symmetries acts transitively on the vertices. In this section, for
each polytope P , we give a construction of a new polytope Ant(P) which generalizes the
construction of the antiprism of a polygon.
308 Ars Math. Contemp. 22 (2022) #P2.08 / 305–315
Let P be an n-polytope. To formally define the antiprism of P , Ant(P), we let P be a
symbol, and define
Ant(P) := {(F,G) | F,G ∈ P, F ≤P G} ∪ {P},
where the order is given by
(F,G) ≤ (H,K) if and only if F ≤P H ≤P K ≤P G; (2.1)
(F,G) ≤ P for every F,G ∈ P. (2.2)
Throughout this section, when we say that an ordered pair of elements of P is an element
of Ant(P), we shall be referring to an element of Ant(P) different than P .
Note then that P is the maximum element of Ant(P) and that, if F−1 and Fn denote
the minimum and maximum elements of P , respectively, then (F−1, Fn) is in fact the
minimum element of Ant(P). Moreover, for H,F,G ∈ P , with H ≤P F ≤P G, we have
that (H,F ) ≤ (F, F ) and (F,G) ≤ (F, F ), but the only element of Ant(P) greater than
(F, F ) is P .
Suppose that P has rank n and its rank function is rankP . Define
rankAnt(P)(F,G) := rank(F,G) = n+ rankP(F )− rankP(G), (2.3)
rankAnt(P)(P ) := rank(P ) = n+ 1.
Note that for every (F,G) ∈ Ant(P), we have that 0 ≤ rankP(G) − rankP(F ) ≤ n +
1, implying that rank(F,G) ∈ {−1, . . . , n} and therefore rank: Ant(P) → {−1, . . . ,
n+ 1}.
Moreover, if rank(F,G) = −1, then
n+ rankP(F )− rankP(G) = −1.
This is equivalent to have that
rankP(G) = n+ 1 + rankP(F ).
But rankP(G) ≤ n which implies that F should have rank −1, and thus G has rank n; in
other words, rank(F,G) = −1 if and only if F = F−1 and G = Fn. We can further see
that rank(F,G) = n if and only if F = G. Hence, the facets of Ant(P) are the elements
(F, F ), with F ∈ P .
There are other faces of Ant(P) that are easy to identify. For example, if (F,G) is a
vertex, it should satisfy that rank(F,G) = n+ rankP(F )− rankP(G) = 0. That is,
rankP(G) = n+ rankP(F ).
Again, since rankP(G) ≤ n we have two options: either rankP(G) = n and rankP(F ) =
0, or rankP(G) = n− 1 and rankP(F ) = F−1. This implies that the vertices of Ant(P)
are either of the form (v, Fn), where v is a vertex of P , or of the form (F−1, f), where f is
a facet of P .
Before showing that Ant(P) is a polytope, let us analyze the case when P is 2-
polytope. Let P be a 2-polytope with vertices {v1, . . . , vp} and edges {e1, . . . ep} in such
a way that for every i = 1, . . . , p− 1, vi, vi+1 ≤ ei, and v1, vp ≤ ep. Let m and M be the
I. Gleason and I. Hubard: The antiprism of an abstract polytope 309
minimum and maximum elements of P , respectively. We already know that Ant(P) has a
unique minimum (m,M) and a unique maximum P , that there are 2p vertices, namely:
(v1,M), . . . , (vp,M), (m, e1), . . . , (m, ep),
and that the facets, 2-faces in this case, are of the form (F, F ), where F is any element of
P . Thus there are 2p+ 2 facets. Finally, the 1-faces are the elements
(e1,M), . . . , (ep,M),
(m, v1), . . . , (m, vp),
(v1, ep), (v1, e1), (v2, e1), . . . , (vp, ep),
and there are 4p of them.
We note that the facets (m,m) and (M,M) are p-gons, as their vertices are of the form
(m, ei) and (vi,M), respectively. In contrast, the facets of type (vi, vi) and (ei, ei) are
triangles, as their only vertices are either of the form (vi,M), (m, ei−1), (m, ei) or of the
form (vi,M), (vi+1,M), (m, ei). It is not too difficult now to see that Ant(P) is in fact
isomorphic to the classical antiprism.
Given an abstract polytope P , we should say that Ant(P) is the antiprism of P .
In order to show that the antiprism of any polytope is again a polytope, we shall start
by analyzing the sections of Ant(P). As we noted before, the only elements of rank
n are of the type (F, F ), where F ∈ P . Let us take a look into the sections QF :=
(F, F )/(F−1, Fn) where, as before, F−1, Fn are the minimum and maximum faces of the
n-polytope P , respectively.
Let us fix a face F of P . If (H,G) ∈ QF , then (H,G) ≤ (F, F ), which implies that
H ≤ F ≤ F ≤ G. In other words, F is a face of the section G/H of P . On the other
hand, if H,G ∈ P are such that H ≤ F and F ≤ G, then (H,G) ∈ QF . That means that
the faces of the section QF are in one to one correspondence with the order pairs (H,G)
of elements of P such that H ≤ F ≤ G.
Since P is a polytope, then P−F := F/F−1 and P
+
F := Fn/F are also polytopes. Let
δ : P+F → (P
+
F )
∗ be a duality mapping P+F to its dual. Now, H ∈ P
−
F if and only if
H ≤ F ; on the other hand, F ≤ G if and only if Gδ ∈ (P+F )∗. Consider now the join
product of P−F with (P
+
F )
∗. We have that
ψ : P−F ⋊⋉ (P
+
F )
∗ → QF (2.4)
(H,Gδ) 7→ (H,G)
is a well-defined bijection between P−F ⋊⋉ (P
+
F )
∗ and QF . Furthermore, note that
(H,Gδ) ≤⋊⋉ (H ′, G′δ) if and only if H ≤ H ′ and Gδ ≤ G′δ, which is equivalent to
have H ≤ H ′ and G′ ≤ G. That is, (H,Gδ) ≤⋊⋉ (H ′, G′δ) if and only if H ≤ H ′ ≤
F ≤ G′ ≤ G, which is equivalent to have that (H,G) ≤QF (H ′, G′). Thus, ψ is an
isomorphism between P−F ⋊⋉ (P
+
F )
∗ and QF . This implies that all the facets of Ant(P)
are abstract polytopes. In particular we note that QF−1 ∼= P∗, while QFn ∼= P .
We turn now our attention to the co-faces P/(F,G) of Ant(P). We observe that
P/(F,G) = {(H,K) ∈ Ant(P) | (F,G) ≤Ant(P) (H,K)} ∪ {P}
∼= {(H,K) ∈ P ∗ P | F ≤P H ≤P K ≤P G)} ∪ {P}
∼= {(H,K) ∈ P ∗ P | H,K ∈ G/F,H ≤G/F K} ∪ {P}
∼= Ant(G/F ).
310 Ars Math. Contemp. 22 (2022) #P2.08 / 305–315
This says that all the co-faces of Ant(P) are antiprisms of polytopes of smaller rank than
that of P .
Theorem 2.1. Let P be an n-polytope, then Ant(P) is an n+ 1 polytope.
Proof. The function given in (2.3) is the desired rank function, with range {−1, . . . , n+1},
and it is clear from the definition that Ant(P) has a minimum and a maximum face.
We now proceed by induction over n.
Let P = {F−1, F0} be a 0-polytope. Then Ant(P) = {(F−1, F0), (F−1, F−1),
(F0, F0), P}, where (F−1, F0) ≤ (F−1, F−1), (F0, F0) ≤ P . Hence, Ant(P) is iso-
morphic to an edge, that is, Ant(P) is a 1-polytope.
Suppose now that the antiprism of any polytope of rank (n − 1) is a polytope and let
P be an n-polytope. Since the facets of Ant(P) are a join product of polytopes, then they
are polytopes. In particular, every flag of Ant(P), when taking away the maximum face,
can be seen to be contained flag of a facet of Ant(P). Since the flags of the facets have all
n+ 2 elements, every flag of Ant(P) has exactly n+ 3 elements.
The diamond condition is satisfied and all the proper sections of Ant(P) are connected:
this is straightforward to see as a proper section of Ant(P) is contained either in a facet or
in a vertex figure of Ant(P). The facets of Ant(P) are joins of polytopes (hence polytopes)
and the vertex figures are anitprisms over proper sections of P , which by hypothesis of
induction are also polytopes.
We only have to see that Ant(P) itself is connected. Let (F,G), (H,K) be two proper
faces of Ant(P). We divide the analysis in several cases, depending on whether or not
F,G,H and K are proper or improper faces of P . Note that F and G (resp. H and
K) cannot be improper face of P simultaneously, unless they are equal. Without loss of
generality, we may assume that rankP(G) ≤ rankP(K).
• If G,K ̸= Fn, then (F,G), (F−1, G), (F−1, F−1), (F−1,K), (H,K) is a sequence
of incident proper faces of Ant(P).
• If F,H ̸= F−1, then (F,G), (F, Fn), (Fn, Fn), (H,Fn), (H,K) is a sequence of
incident proper faces of Ant(P).
• If K = Fn, F = F−1 and H is a proper face of P , then H ̸= F−1, Fn and G ̸=
Fn. Since P is connected, then there exists a sequence G = G1, G2, . . . , Gh = H
of incident faces of P all of which, except perhaps for G, are proper faces. Then
(F,G) = (F−1, G1), (F−1, G2), . . . , (F−1, Gh) = (F−1, H), (H,H), (H,Fn) =
(H,K) is a sequence of incident proper faces of Ant(P).
• If K = H = Fn, F = F−1 and G is a proper face of P , then (F−1, G), (G,G),
(G,Fn), (Fn, Fn) is a sequence of incident proper faces of Ant(P).
• If K = H = Fn, F = G = F−1, then let J ∈ P be any proper face of P (exists as
we are assuming n > 0). Hence (F−1, F−1), (F−1, J), (J, J), (J, Fn), (Fn, Fn) is
a sequence of incident proper faces of Ant(P).
Hence Ant(P) is connected and therefore it is a polytope.
Note that given a polytope P and its dual P∗, there is a duality δ : P → P∗. We know
that δ is a bijection that reverses the order, and hence every element of P∗ can be written
as Fδ, where F is a face of P . Hence, there is a natural bijection between the faces of
I. Gleason and I. Hubard: The antiprism of an abstract polytope 311
the antirpism of P and the faces of the antiprism of P∗. In fact, we have the following
proposition.
Proposition 2.2. For any polytope P , Ant(P) ∼= Ant(P∗), where P∗ denotes the dual
of P .
Proof. Let P and P ∗ be the maximum elements of Ant(P) and Ant(P∗), respectively,
and let δ : P → P∗ be a duality. Let ψ : Ant(P) → Ant(P∗) be given by:
(F,G) 7→ (Gδ, Fδ),
P 7→ P ∗.
Then clearly ψ is a well-defined bijection between Ant(P) and Ant(P∗). Furthermore
(F,G) ≤Ant(P) (H,K) if and only if F ≤P H ≤P K ≤P G if and only if Gδ ≤P∗
Kδ ≤P∗ Hδ ≤P∗ Fδ if and only if (Gδ, Fδ) ≤P∗ (Kδ,Hδ) which is equivalent to
(F,G)ψ ≤P∗ (H,K)ψ. Since it is now straightforward to see that δ−1 also induces a
bijection that preserves the order and is the inverse of ψ. This settles the proposition.
2.1 The flags of a polytope and the flags of its antiprism
In this section we study the relation between the flags of Ant(P) and the flags of P .
Let P be an n-polytope (with maximum element Fn and minimum element F−1) and
consider V to be the set of all ordered (n + 1)-tuples with entries 0 and 1. We are going
to see that there is a bijection between F(Ant(P)) and F(P) × V . For this, consider a
flag of Ant(P), {A−1, A0, . . . An+1}, where rankAi = i. Then An+1 = P and for each
i = −1, 0 . . . n, there exist F i, Gi ∈ P such that Ai = (F i, Gi). It is straightforward to
see that F−1 = F−1, G−1 = Fn and Fn = Gn := F , for some F ∈ P . Furthermore,
observe that
F−1 ≤ F 0 ≤ F 1 ≤ · · · ≤ Fn = F = Gn ≤ Gn−1 ≤ · · · ≤ G0 ≤ G−1 = Fn, (2.5)
is a sequence of faces of P in which, of course, many of the elements might repeat. For
example, the sequence could be such that F 0 = F 1 = F 2 = · · · = Fn = F−1.
On one hand, note that for a given i ∈ {0, . . . n}, we have that either
rank(F i) = rank(F i+1) and rank(Gi) = rank(Gi+1) + 1
or
rank(F i) + 1 = rank(F i+1) and rank(Gi) = rank(Gi+1).
In particular, either rank(F 0) = −1 and rank(G0) = n − 1 or rank(F 0) = 0 and
rank(G0) = n. Hence, we can regard the sequence in (2.5) as a sequence of incident
faces of P that has exactly one element of each rank. That is, a flag of P . In other words,
each flag of Ant(P) induces a flag of P in a natural way.
On the other hand, the sequence in (2.5) also defines an element of V in the following
way. For each i ∈ {0, . . . , n}, let ai = 0 if rank(F i−1) = rank(F i) and ai = 1 otherwise.
It should be clear that (a0, . . . an) is an element of V .
The above assignment is a bijection. To see this, take Φ ∈ F(P) and v ∈ V . Denote
by Φi the i-face of Φ and by vi the i-th element of v, i.e. v = (v0, v1, . . . , vn). We define
the flag {A−1, A0, . . . An+1} of Ant(P), where eachAi = (F i, Gi), in the following way.
312 Ars Math. Contemp. 22 (2022) #P2.08 / 305–315
First, An+1 = P and A−1 = (F−1, Fn) = (Φ−1,Φn). Now, we define inductively the
elements F i and Gi. Suppose F i−1 is defined as the j-face Φj , then we define F i :=
Φj+vi . (For example, if v0 = 0, then F
0 = Φ−1+v0 = Φ−1 = F−1, and if v0 = 1,
then F 0 = Φ−1+v0 = Φ0.) Similarly, we first define G
n := Fn and, inductively, suppose
that Gi+1 is defined as the k-face Φk, then we define Gi := Φk+1−vi+1 . Thus, | {vj ∈
v | vj = 1} |= m + 1, for some −1 ≤ m ≤ n and hence Gn = Fn = Φm and
| {vj ∈ v | vj = 0} |= n−m. This implies that
G−1 = Φm+(1−vn)+(1−vn−1+···+(1−v0)) = Φm+(n−m) = Φn.
It should not be difficult to see that this assignment of a flag of Ant(P), given a pair
(Φ, v) ∈ F(P)×V is inverse to the above description, where each flag of Ant(P) induces
a flag of P and an element of V . We have therefore established that
Lemma 2.3. Let P be an n-polytope. Then the flags of Ant(P) are in one-to-one corre-
spondence with the set F(P)× V , where F(P) denotes the set of flags of P and V the set
of all ordered (n+ 1)-tuples with entries 0 and 1.
3 Products and the antiprism
In the next section we will study the automorphism group of an antiprism. We shall see that
computing it for polytopes that are prime with respect to the join product is straighforward.
To completely determine the automorphism group of any antiprism, we need some of the
results given in this section.
All our results here deal with the interaction of the join product and the direct sum with
the anitprism.
Proposition 3.1. Let P and Q be two polytopes. Then Ant(P ⋊⋉ Q) ∼= Ant(P)⊕Ant(Q).
Proof. Let ψ : Ant(P ⋊⋉ Q) → Ant(P)⊕Ant(Q) be such that(
(F,G), (H,K))ψ =
(
(F,H), (G,K)
)
and, if P, PP and PQ are the maximum elements of Ant(P ⋊⋉ Q),Ant(P) and Ant(Q),
respectively, then Pψ = (PP , PQ). We shall show that ψ is an isomorphism.
First note that
(
(F,G), (H,K)
)
∈ Ant(P ⋊⋉ Q) implies that (F,G), (H,K) ∈ P ⋊⋉ Q
and that (F,G) ≤P⋊⋉Q (H,K). Hence, we have that F,H ∈ P with F ≤P H , and that
G,K ∈ Q with G ≤Q K; that is, (F,H) ∈ Ant(P) and (G,K) ∈ Ant(Q). Moreover,
(F,G) is not the maximum element of Ant(P), and (H,K) is not the maximum element of
Ant(Q), which implies that
(
(F,H), (G,K)
)
∈ Ant(P)⊕Ant(Q). Furthermore, observe
that different elements of Ant(P ⋊⋉ Q) go to different elements of Ant(P)⊕Ant(Q) under
ψ and therefore ψ is a well-defined function from Ant(P ⋊⋉ Q) to Ant(P)⊕Ant(Q).
Similarly, let ϕ : Ant(P)⊕Ant(Q) → Ant(P ⋊⋉ Q) be such that(
(F,H), (G,K)
)
7→
(
(F,G), (H,K)
)
.
A similar argument as the one above shows that ϕ is also a well-defined function. Note that
both ψϕ and ϕψ are the identity map, which implies that both functions are bijections, and
one is the inverse of the other.
I. Gleason and I. Hubard: The antiprism of an abstract polytope 313
We need to show that these two functions preserve the orders. Let
(
(F0, G0), (H0,K0)
)
,(
(F1, G1), (H1,K1)
)
∈ Ant(P ⋊⋉ Q), then(
(F0, G0), (H0,K0)
)
≤Ant(P⋊⋉Q)
(
(F1, G1), (H1,K1)
)
⇔ (F0, G0) ≤P⋊⋉Q (F1, G1) ≤P⋊⋉Q (H1,K1) ≤P⋊⋉Q (H0,K0)
⇔ F0 ≤P F1 ≤P H1 ≤P H0 and G0 ≤Q G1 ≤Q K1 ≤Q K0
⇔ (F0, H0) ≤Ant(P) (F1, H1) and (G0,K0) ≤Ant(Q) (G1,K1)
⇔
(
(F0, H0), (G0,K0)
)
≤Ant(P)⊕Ant(Q)
(
(F1, H1), (G1,K1)
)
.
Therefore both ψ and ϕ preserve the orders and hence ψ is an isomorphism.
Lemma 3.2. If P is a prime polytope with respect to the join product, then Ant(P) is a
prime polytope with respect to the direct sum.
Proof. Suppose otherwise. Then there exists a polytope P that is prime with respect to
the join product, but such that Ant(P) is not prime with respect to the direct sum. Let
Ant(P) = Q⊕K, where Q and K are polytopes of rank at least 1.
Note that Ant(P) contains a facet that is isomorphic to P . In fact, if Fn denotes the
maximum element of P , then (Fn, Fn) ∈ Ant(P) has rank n and if (F,G) ≤ (Fn, Fn),
then G = Fn (since F ≤P Fn ≤P Fn ≤P G). That means that the section
(Fn, Fn)/(F−1, Fn) of Ant(P) is isomorphic to P .
But by Lemma 1.1, a facet of the direct product Q ⊕ K is isomorphic to a non-trivial
join product. Hence P is not prime with respect to the join product, which contradicts our
hypothesis.
4 Automorphism groups
We now turn our attention to the study of the automorphism group of the antiprism of P .
It is not difficult to see that every automorphism of P induces an automorphism of
Ant(P). In fact, given γ ∈ Γ(P) the mapping γ̂ : Ant(P) → Ant(P) given by (F,G)γ̂ :=
(Fγ,Gγ), for (F,G) ∈ Ant(P), and P γ̂ := P is clearly an automorphism of Ant(P).
Similarly, if P is a self-dual polytope and δ is a duality of P , then δ̂ : (F,G) 7→ (Gδ, Fδ)
(and P δ̂ = P ) is also an automorphism of Ant(P). In other words, we have the following
lemma. Keep in mind that we have defined the extended group of a non-self-dual polytope
simply as its automorphism group.
Lemma 4.1. Let P be a polytope and let Γ̂(P) denote its extended group. Then, Γ̂(P) is
(isomorphic to) a subgroup of G(Ant(P)).
It is not difficult to see that if ψ : F(Ant(P)) → F(P) × V is the bijection from
Lemma 2.3, γ ∈ Γ̂(P) and γ̃ is the automorphism of Ant(P) induced by γ, then for every
flag Φ ∈ F(P) and every (n+1)-tuple v ∈ V , we have that (Φ, v)ψ−1γ̃ψ = (Φγ, v). This
implies that if P is a self-dual polytope, dualities of P induce automorphisms of Ant(P).
The above observation, together with Lemmas 4.1 and 2.3 imply the following result.
Proposition 4.2. Let P an n-polytope and let Ant(P) be its antiprism. If P is a k-orbit
polytope, and Ant(P) is an m-orbit polytope, then:
• if P is self-dual, then m ≤ k · 2n,
314 Ars Math. Contemp. 22 (2022) #P2.08 / 305–315
• if P is not self-dual, then m ≤ k · 2n+1.
Observe that, by the isomorphism given in (2.4), the facets of Ant(P) can be seen as
the join product of sections of P . This means that, maybe with the exception of the facets
(F−1, F−1) ∼= P∗ and (Fn, Fn) ∼= P , the facets of Ant(P) are not prime with respect to
the join product. Whenever P is a prime polytope with respect to the join product, we can
obtain a lot of information about Γ(Ant(P).
Proposition 4.3. Let P be a prime polytope with respect to the join product. Then,
Γ(Ant(P)) ∼= Γ̂(P).
Proof. By Lemma 4.1 we only need to show that any automorphism of Ant(P) is in fact
induced by either an automorphism or a duality (if P is self-dual) of P . In this proof we
abuse notation and refer to a polytope that is prime with respect to the join product simply
as a prime polytope.
As pointed out above, when P is a prime polytope, the only two facets of Ant(P)
that are also prime are (F−1, F−1) and (Fn, Fn). This means that any automorphism α ∈
Γ(Ant(P)) either fixes both such faces or interchanges them (as they cannot be permuted
with any other, or they would not be prime). It is then easy to see that if α fixes them, then
it induces an automorphism of P and that if interchanges them, then P is self-dual and α
induces a duality.
Propositions 4.2 and 4.3 immediately imply the following result.
Corollary 4.4. Let P an n-polytope that is prime with respect to the join product and let
Ant(P) be its antiprism. If P is a k-orbit polytope, and Ant(P) is an m-orbit polytope,
then:
• if P is self-dual, then m = k · 2n,
• if P is not self-dual, then m = k · 2n+1.
Lemma 3.2, together with Propositions 3.1 and 4.3 and Theorem 1.2, give us all the
necessary tools to compute the automorphism of the antiprism of any polytope.
Theorem 4.5. Let P = Qm11 ⋊⋉ Q
m2
2 ⋊⋉ · · · ⋊⋉ Qmrr , where each Qi is a prime polytope
with respect to the join product. Then
Γ(Ant(P)) = Πri=1
(
(Γ̂(Qi))mi ⋊ Smi
)
.
ORCID iDs
Isabel Hubard https://orcid.org/0000-0002-0960-3671
References
[1] A. Björner, The antiprism fan of a convex polytope, Am. Math. Soc. 18 (1997).
[2] M. N. Broadie, A theorem about antiprisms, Linear Algebra Appl. 66 (1985), 99–111, doi:10.
1016/0024-3795(85)90127-2.
[3] M. Dobbins, Antiprismless, or: Reducing combinatorial equivalence to projective equivalence
in realizability problems for polytopes, 2013, arXiv:1307.0071 [math.CO].
I. Gleason and I. Hubard: The antiprism of an abstract polytope 315
[4] I. Gleason and I. Hubard, Products of abstract polytopes, J. Comb. Theory, Ser. A 157 (2018),
287–320, doi:10.1016/j.jcta.2018.02.002.
[5] P. McMullen and E. Schulte, Abstract Regular Polytopes, volume 92 of Encycl. Math. Appl.,
Cambridge University Press,Cambridge, 2002, doi:10.1017/cbo9780511546686.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.09 / 317–326
https://doi.org/10.26493/1855-3974.2577.25d
(Also available at http://amc-journal.eu)
Linkedness of Cartesian products of complete
graphs*
Leif K. Jørgensen
Department of Mathematical Sciences, Aalborg University, Denmark
Guillermo Pineda-Villavicencio † , Julien Ugon ‡
Federation University, Ballarat, Australia and
School of Information Technology, Deakin University, Geelong, Australia
Received 9 March 2021, accepted 26 August 2021, published online 27 May 2022
Abstract
This paper is concerned with the linkedness of Cartesian products of complete graphs.
A graph with at least 2k vertices is k-linked if, for every set of 2k distinct vertices organised
in arbitrary k pairs of vertices, there are k vertex-disjoint paths joining the vertices in the
pairs.
We show that the Cartesian product Kd1+1 × Kd2+1 of complete graphs Kd1+1 and
Kd2+1 is ⌊(d1 + d2)/2⌋-linked for d1, d2 ≥ 2, and this is best possible.
This result is connected to graphs of simple polytopes. The Cartesian product
Kd1+1 ×Kd2+1 is the graph of the Cartesian product T (d1)× T (d2) of a d1-dimensional
simplex T (d1) and a d2-dimensional simplex T (d2). And the polytope T (d1) × T (d2) is
a simple polytope, a (d1 + d2)-dimensional polytope in which every vertex is incident to
exactly d1 + d2 edges.
While not every d-polytope is ⌊d/2⌋-linked, it may be conjectured that every simple d-
polytope is. Our result implies the veracity of the revised conjecture for Cartesian products
of two simplices.
Keywords: k-linked, cyclic polytope, connectivity, dual polytope, linkedness, Cartesian product.
Math. Subj. Class. (2020): 05C40, 52B05
*The authors want to thank the referee for his/her comments, which have certainly helped to improve the
presentation of the paper.
†Corresponding author. Guillermo would like to thank the hospitality of Leif Jørgensen, and the Department
of Mathematical Sciences at Aalborg University, where this research started.
‡Julien Ugon’s research was supported by the ARC discovery project DP180100602.
E-mail addresses: leifkjorgensen@gmail.com (Leif K. Jørgensen), work@guillermo.com.au (Guillermo
Pineda-Villavicencio), julien.ugon@deakin.edu.au (Julien Ugon)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
318 Ars Math. Contemp. 22 (2022) #P2.09 / 317–326
1 Introduction
Denote by V (X) the vertex set of a graph. Given sets A,B of vertices in a graph, a path
from A to B, called an A−B path, is a (vertex-edge) path L := u0 . . . un in the graph such
that V (L) ∩ A = {u0} and V (L) ∩ B = {un}. We write a − B path instead of {a} − B
path, and likewise, write A− b path instead of A− {b}.
Let G be a graph and X a subset of 2k distinct vertices of G. The elements of X
are called terminals. Let Y := {{s1, t1}, . . . , {sk, tk}} be an arbitrary labelling and (un-
ordered) pairing of all the vertices in X . We say that Y is linked in G if we can find disjoint
si − ti paths for i ∈ [1, k], the interval 1, . . . , k. The set X is linked in G if every such
pairing of its vertices is linked in G. Throughout this paper, by a set of disjoint paths, we
mean a set of vertex-disjoint paths. If G has at least 2k vertices and every set of exactly 2k
vertices is linked in G, we say that G is k-linked.
This paper studies the linkedness of Cartesian products of complete graphs. Linkedness
of Cartesian products has been studied in the past [4]. The Cartesian product G1 ×G2 of
two graphs G1 and G2 is the graph defined on the pairs (v1, v2) with vi ∈ Gi and with
two pairs (u1, u2) and (v1, v2) being adjacent if, for some ℓ ∈ {1, 2}, uℓvℓ ∈ E(Gℓ) and
ui = vi for i ̸= ℓ. We prove that the Cartesian product Kd1+1×Kd2+1 of complete graphs
Kd1+1 and Kd2+1 is ⌊(d1 + d2)/2⌋-linked for d1, d2 ≥ 0, and that there are products that
are not ⌊(d1 + d2 + 1)/2⌋-linked; hence this result is best possible. Here Kt denotes the
complete graph on t vertices.
Our result is connected to questions on the linkedness of a polytope. A (convex) poly-
tope is the convex hull of a finite set X of points in Rd; the convex hull of X is the smallest
convex set containing X . The dimension of a polytope in Rd is one less than the maximum
number of affinely independent points in the polytope; a set of points p⃗1, . . . , p⃗k in Rd is
affinely independent if the k − 1 vectors p⃗1 − p⃗k, . . . , p⃗k−1 − p⃗k are linearly independent.
A polytope of dimension d is referred to as a d-polytope.
The Cartesian product P × P ′ of a d-polytope P ⊂ Rd and a d′-polytope P ′ ⊂ Rd′ is
the Cartesian product of the sets P and P ′:
P × P ′ =
{(
p
p′
)
∈ Rd+d
′
∣∣∣∣ p ∈ P, p′ ∈ P} .
The resulting polytope is (d + d′)-dimensional. The graph G(P ) of a polytope P is the
undirected graph formed by the vertices and edges of the polytope. It follows that the graph
G(P ×P ′) of the Cartesian product P ×P ′ is the Cartesian product G(P )×G(P ′) of the
graphs G(P ) and G(P ′).
A d-simplex T (d) is the convex hull of d + 1 affinely independent points in Rd. The
graph of T (d) is the complete graph Kd+1. As a consequence, our result implies that the
graph of the Cartesian product T (d1) × T (d2) is ⌊(d1 + d2)/2⌋-linked for d1, d2 ≥ 0.
Henceforth, if the graph of a polytope is k-linked we say that the polytope is also k-linked.
The first edition of the Handbook of Discrete and Computational Geometry [3, Prob-
lem 17.2.6] posed the question of whether or not every d-polytope is ⌊d/2⌋-linked. This
question was answered in the negative by [2]. None of the known counterexamples are sim-
ple d-polytopes, d-polytopes in which every vertex is incident to exactly d edges. Hence, it
may be hypothesised that the conjecture holds for such polytopes.
Conjecture 1.1. Every simple d-polytope is ⌊d/2⌋-linked for d ≥ 2.
L. K. Jørgensen et al.: Linkedness of Cartesian products of complete graphs 319
Cartesian products of simplices are simple polytopes, and so our result supports this
revised conjecture. Furthermore, Cartesian products of simplices and duals of cyclic poly-
topes are related; the dual of a cyclic d-polytope with d+2 vertices is the Cartesian product
of a ⌊d/2⌋-simplex and a ⌈d/2⌉-simplex [6, Example 0.6]. Hence we obtain that the dual
of a cyclic d-polytope on d+ 2 vertices is also ⌊d/2⌋-linked for d ≥ 2.
Unless otherwise stated, the graph theoretical notation and terminology follows from
[1] and the polytope theoretical notation and terminology from [6]. Moreover, when refer-
ring to graph-theoretical properties of a polytope such as linkedness and connectivity, we
mean properties of its graph.
2 Linkedness of Cartesian products of complex graphs
The contribution of this section is a sharp theorem (Theorem 2.1) that tells the story of the
linkedness of Cartesian product of two complete graphs.
(e)
s1
t2
s2
t1
s3 t3 sk tk· · ·
· · ·
(a)
s1
s2
s3t2
t3
t1
s1
s2
s3
t2
t3
t1
s4
t4
s1
s2
s3
t5
s5
s6
s4
t4
t6
t1t2
t3
(b)
(c)
s1
s2
s3t5
s5
t1
s4
t3
(d)
t4
t2
Figure 1: No feasible linkage problems for Kd1+1 × Kd2+1, k = ⌊(d1 + d2 + 1)/2⌋,
d1 ≤ 2 and d2 > d1. (a) The case d1 = 1 and even d2 with d2 > d1. (b) The case d1 = 2
and d2 = 3. (c) The case d1 = 2 and d2 = 5. (d) The case d1 = 2 and d2 = 7. (e) The
case d1 = 2 and d2 = 9. Each row of each part (a)-(e) is a complete graph whose edges
have not been drawn.
Theorem 2.1. The Cartesian product of two complete graphs Kd1+1 and Kd2+1 is
⌊(d1 + d2)/2⌋-linked for every d1, d2 ≥ 0.
Remark 2.2. Theorem 2.1 is best possible. There are products Kd1+1 × Kd2+1 that are
not ⌊(d1 + d2 + 1)/2⌋-linked:
1. K2 ×Kd2+1 for even d2 ≥ 1, and
320 Ars Math. Contemp. 22 (2022) #P2.09 / 317–326
2. K3 ×Kd2+1 for d2 = 1, 3, 5, 7, 9.
For each of these cases, Figure 1 provides a pairing of terminals that cannot be
⌊(d1 + d2 + 1)/2⌋-linked. We conjecture these are the only such cases.
An immediate corollary of Theorem 2.1 is the following.
Corollary 2.3. The Cartesian product of two simplices T (d1) and T (d2) is ⌊(d1+d2)/2⌋-
linked for every d1, d2 ≥ 0.
The notions of linkage, linkage problem, and valid path will simplify our arguments.
A linkage in a graph is a subgraph in which every component is a path. Let X be a set of
vertices in a graph and let Y := {{s1, t1} , . . . , {sk, tk}} be a pairing of all the vertices of
X . A Y -linkage {L1, . . . , Lk} is a set of disjoint paths with the path Li joining the pair
{si, ti} for i = 1, . . . , k. We may also say that Y represents our linkage problem, and if Y
is linked in G then our linkage problem is feasible and infeasible otherwise. A path in the
graph is called X-valid if no inner vertex of the path is in X . Let X be a set of vertices
in a graph G. Denote by G[X] the subgraph of G induced by X , the subgraph of G that
contains all the edges of G with vertices in X . Write G−X for G[V (G) \X].
Consider a linkage problem Y := {{s1, t1}, . . . , {sk, tk}} on a set X of 2k vertices
in a graph G. Consider a linkage L from a subset Z of X to some set Z ′ disjoint from
X and label the vertices of Z ′ such that the path in L with end zi ∈ Z has its other end
z′i ∈ Z ′. Then the linkage L in G induces a linkage problem Y ′ in (G−V (L))∪Z ′ where
the vertices of X \ Z remain and the vertices of Z have been replaced by the vertices of
Z ′. Slightly abusing terminology, we also call terminals the vertices of Z ′. If the problem
Y ′ is feasible in (G− V (L)) ∪ Z ′, so is the problem Y in G.
Since we make heavy use of Menger’s theorem [1, Theorem. 3.3.1], we next remind
the reader of one of its consequences.
Theorem 2.4 (Menger’s theorem). Let G be a k-connected graph, and let A and B be two
subsets of its vertices, each of cardinality at least k. Then there are k disjoint A−B paths
in G.
We fix some notation and terminology for the remaining of the section. Let G denote
the graph Kd1+1 ×Kd2+1. We think of G = Kd1+1 ×Kd2+1 as a grid with d1 + 1 rows
and d2 + 1 columns. In this way, the entry in Row i and Column j can be referred to as
G[i, j].
When we write about a row r of subgraph G′ of G, we think of r as a subgraph of G′
and as the number r so that we can write about the rth row of G′ or G; this ambiguity
should cause no confusion. An entry in the grid Kd1+1 ×Kd2+1 with no terminal is said
to be free, as is a row or a column of a subgraph of G with no terminal. A row or a column
of a subgraph of G with every entry being occupied by a terminal is said to be full.
We need the following induced subgraphs of G:
Cab...z, the subgraph formed by the union of Columns a, b, . . . , z;
C̄ab...z, the subgraph obtained by removing Columns a, b, . . . , z;
Rab...z, the subgraph formed by the union of Rows a, b, . . . , z;
R̄ab...z, the subgraph obtained by removing Rows a, b, . . . , z;
Aα, the induced subgraph of C̄12 obtained by removing its first α rows; and
Bα, the subgraph of C12 obtained by removing its first α rows.
L. K. Jørgensen et al.: Linkedness of Cartesian products of complete graphs 321
For instance, C̄1 denotes the subgraph of G obtained by removing the first column, C12
the subgraph formed by the first two columns of G, and C̄12 denotes the subgraph obtained
by removing the first two columns of G; observe C̄12 is isomorphic to Kd1+1 × Kd2−1.
Figure 2 depicts some of the aforementioned subgraphs of Kd1+1 ×Kd2+1.
...
...
...
...
...
...
...
...
...
...
AαBα
 α rows d1 + 1 – α rows︸ ︷︷ ︸
C12
︸ ︷︷ ︸
C̄12
Figure 2: Depiction of the subgraphs Bα, Aα, C12, and C̄12 of Kd1+1 ×Kd2+1.
The connectivity of Kd1+1 ×Kd2+1 is stated below.
Lemma 2.5 (Špacapan [5, Theorem 1]). The (vertex)connectivity of Kd1+1 × Kd2+1 is
precisely d1 + d2.
We continue fixing further notation. Henceforth let k := ⌊(d1 + d2)/2⌋. And let X be
a subset of 2k vertices of G and let Y := {{s1, t1}, . . . , {sk, tk}} be a pairing of all the
vertices in X .
We first settle the simple cases of (0, d2) and (1, d2) for d2 ≥ 0.
Proposition 2.6 (Base cases). For d2 ≥ 0 the Cartesian products K1 × Kd2+1 and
K2 ×Kd2+1 are both ⌊(1 + d2)/2⌋-linked. This statement is best possible.
Proof. The lemma is true for the pair (0, d2) for each d2 ≥ 0, since K1×Kd2+1 = Kd2+1
and Kd2+1 is ⌊(1 + d2)/2⌋-linked. This is best possible.
The graph K2 × Kd2+1 is (1 + d2)-connected by Lemma 2.5. Use Menger’s theo-
rem (Theorem 2.4) to bring the 1 + d2 terminals to the subgraph R̄1 through a linkage
{S1, . . . , Sk, T1, . . . , Tk} with Si := si − R̄1 and Ti := ti − R̄1 for i ∈ [1, k]. Letting
{s̄i} := V (Si) ∩ V (R̄1) and {t̄i} := V (Ti) ∩ V (R̄1), we produce a new linkage prob-
lem Y ′ := {{s̄1, t̄1}, . . . , {s̄k, t̄k}} in R̄1 whose feasibility implies that of Y in G. To
solve Y ′ link the pairs of Y ′ in the subgraph R̄1, which is isomorphic to Kd2+1, using
the ⌊(1 + d2)/2⌋-linkedness of Kd2+1. For even even d2, Figure 1(a) shows an infeasible
linkage problem with ⌊(2 + d2)/2⌋ pairs in the graph K2 ×Kd2+1.
In what follows we aim to find a Y -linkage {L1, . . . , Lk} in G with Li joining the pair
{si, ti} of Y for i ∈ [1, k]. Our proof is by induction on (d1, d2) with the base cases settled
in Proposition 2.6. If there is a pair of Y , say {s1, t1}, lying in some column or row of
G, say in Column 1, we send every terminal si ∈ C1 that is different from s1 and t1 and
that is not adjacent to ti to the subgraph C̄1, and apply the induction hypothesis on C̄1.
Otherwise, we may assume every pair of Y lies in two distinct columns or rows, say the
pair {s1, t1} lies in C12; then we send every terminal si ∈ C12 that is different from s1 and
322 Ars Math. Contemp. 22 (2022) #P2.09 / 317–326
t1 and that is not adjacent to ti to the subgraph C̄12, and apply the induction hypothesis to
C̄12. We develop these ideas below.
The definition of k-linkedness gives the following lemma at once; we will use it im-
plicitly hereafter.
Lemma 2.7. Let ℓ ≤ k. Let X be a set of 2ℓ distinct vertices of a k-linked graph K, let Y
be a labelling and pairing of the vertices in X , and let Z be a set of 2k − 2ℓ vertices in K
such that X ∩ Z = ∅. Then there exists a Y -linkage in K that avoids every vertex in Z.
Besides, basic algebraic manipulation yields the following inequality.
Lemma 2.8. If x ≥ 2 and y ≥ 2 then x(y − 1) > x+ y − 3.
Proof. The inequality simplifies to (x− 1)(y − 2) > −1.
We are now ready to put together all the elements of the proof of Theorem 2.1.
Proof of Theorem 2.1. Let k := ⌊(d1 + d2)/2⌋. Then d1 + d2 ≥ 2k.
Proposition 2.6 gives the result for the pairs (d1, 0), (0, d2), (d1, 1), and (1, d2) for each
d1, d2 ≥ 0. Hence, our bidimensional induction on (d1, d2) can start with the assumption
of d1, d2 ≥ 2.
We first deal with the case where a pair in Y , say {s1, t1}, lies in some column or some
row of G, say in Column 1.
Case 1. A pair in Y , say {s1, t1}, lies in Column 1.
The induction hypothesis ensures that the subgraph C̄1 is (k − 1)-linked. Hence it
suffices to show that all the terminals in C1 other than s1, t1 can be moved to C̄1 via a
linkage; Menger’s theorem (Theorem 2.4) guarantees this.
Let U be the set of terminals in C1 other than s1 and t1, and let W be the set of terminals
in C̄1. Then |U |+|W | ≤ d1+d2−2, as |U |+|W | = 2k−2 and 2k ≤ d1+d2. Besides, the
subgraph G− (W ∪{s1, t1}) is |U |-connected, as G is (d1 + d2)-connected (Lemma 2.5).
In the case of d1, d2 ≥ 2, Lemma 2.8 yields that C̄1 has more than |U ∪W | vertices:
|C̄1| = (d1 + 1)d2 > d1 + 1 + d2 + 1− 3 > d1 + d2 − 2 = |U |+ |W |.
Use Menger’s theorem (Theorem 2.4) to bring the |U | terminals in C1 to the subgraph
C̄1 through a linkage YU . For every path L in YU , if si ∈ L, let {s̄i} := V (L) ∩ V (C̄1)
and if ti ∈ L let {t̄i} := V (L)∩V (C̄1). For si ∈ W (respectively ti ∈ W ) let s̄i = si (re-
spectively t̄i = ti). This produces a new linkage problem Y ′ := {{s̄2, t̄2}, . . . , {s̄k, t̄k}}
in C̄1 whose feasibility implies that of Y in G, since s1 and t1 are adjacent in C1. The
(k − 1)-linkedness of C̄1 now settles the case.
By symmetry, we can assume that every pair {si, ti} in Y lies in two different columns
or rows and that si, ti are not adjacent. Without loss of generality, assume that
s1 is in Column 1 and t1 is in Column 2 of C12. (∗)
The induction hypothesis also ensures that both C̄12 and R̄12 are (k − 1)-linked. We
consider two further cases based on the number of terminals in C12 or R12.
Case 2. The subgraph C12 contains precisely d1 + 2 − α terminals, including {s1, t1},
where 0 ≤ α ≤ d1.
L. K. Jørgensen et al.: Linkedness of Cartesian products of complete graphs 323
Excluding {s1, t1}, there are at most d1 terminals in C12, and there are d1+1 internally-
disjoint s1 − t1 paths in C12 of length at most three: two length-two paths and d1 − 1
length-three paths. One of these s1 − t1 paths, say L1, avoids every other terminal in C12.
Without loss of generality, assume that Row 1 in C12 is part of the path L1; that is,
{G[1, 1], G[1, 2]} ⊆ V (L1). (∗∗)
In the subcase α = d1, every pair in Y \{s1, t1} is in C̄12, and the induction hypothesis
on C̄12 settles the subcase.
Suppose that α = d1 − 1, say C12 contains {s1, t1, s2}. Then s2 ∈ B1 and t2 ∈ C̄12.
We may assume s1, s2 are in Column 1 and t1 is in Column 2. We show there is an X-valid
s2 −A1 path L′2 such that the vertex s̄2 ∈ V (L′2) ∩ V (A1) is either t2 or a nonterminal.
Through each entry of Column 1 of B1, there are d2 − 1 paths form s2 to A1 of length
at most two (one for each column in A1). Moreover, there are at least d1 − 1 free entries
in Column 1 of B1. Therefore, to ensure the existence of L′2, we need to show that at least
one of these (d1 − 1)(d2 − 1) paths from s2 to A1 either contains t2 or a nonterminal in
A1. Indeed, according to Lemma 2.8, the inequality
(d1 − 1)(d2 − 1) > d1 − 1 + d2 − 3 ≥ |X \ {s1, t1, s2, t2}|
holds for d1, d2 ≥ 2. Hence we get the existence of L′2. As a result, the solution of the new
problem Y ′ := {{s̄2, t2} , {s3, t3} , . . . , {sk, tk}} in C̄12 induces a solution of the problem
Y in G. And the solution of Y ′ follows from the (k − 1)-linkedness of C̄12.
Henceforth assume that α ≤ d1 − 2. To finalise Case 2, we require a couple of claims.
Claim 2.9. Suppose that there are at most d1 +2−α terminals in Bα+1 = Kd1−α ×K2.
Then there is an injection from the set of rows of Bα+1 that contain two terminals x1, x2
such that {x1, x2}∩ {s1, t1} = ∅ to the set of rows of Bα+1 that contain no terminal other
than possibly s1 and t1.
Proof. This follows from a simple counting argument. The number of rows in Bα+1 is
d1 − α. Let m denote the number of rows of Bα+1 that contain two terminals x1, x2 such
that {x1, x2}∩{s1, t1} = ∅ and let n := |(X ∩V (Bα+1)) \ {s1, t1} |; that is, n counts the
total number of terminals in Bα+1 other than s1 and t1. It follows that the number of rows
of Bα+1 that contain precisely one terminal x ̸∈ {s1, t1} is n− 2m; either s1 or t1 may be
in these rows. As a result, the number of rows of Bα+1 that contain no terminal other than
{s1, t1} is d1 − α −m − (n − 2m). Combining n ≤ d1 − α with all these numbers, we
get that
d1 − α−m− (n− 2m) = d1 − α− n+m ≥ d1 − α− (d1 − α) +m = m.
The claim is proved.
Claim 2.10. Suppose that there are at most d1+2−α terminals in Bα+1 = Kd1−α×K2.
If every row in the subgraph Aα+1 = Kd1−α ×Kd2−1 of C̄12 has a free entry, then, for
every terminal x ̸∈ {s1, t1} in Bα+1, there is an X-valid x− Aα+1 path L to a free entry
in Aα+1; and all these X-valid paths are disjoint.
Proof. If a row of Bα+1 contains exactly one terminal x ̸∈ {s1, t1}, then send x to a
free entry in the same row of Aα+1. Let x1 and x2 be two terminals in Bα+1 that satisfy
324 Ars Math. Contemp. 22 (2022) #P2.09 / 317–326
{x1, x2}∩{s1, t1} = ∅ and occupy a row rf of Bα+1. From Claim 2.9 ensues the existence
of a row re of Bα+1 that contain no terminal other than possibly s1 and t1; in short, there
is at least a free entry in re.
Consider a pair (rf , re) of rows granted by Claim 2.9. Send either x1 or x2, say x1,
to the free entry in the row re of Aα+1 passing through the corresponding free entry in the
row re of Bα+1, and send x2 to a free entry in the row rf of Aα+1. The proof of the claim
is now complete.
Now suppose that α = 0 or 2 ≤ α ≤ d1 − 2. In this subcase, the subgraph C̄12
contains at most α full rows: if α + 1 rows were full in C̄12 then there would be at least
(α+1)(d2−1) terminals in C̄12 but (α+1)(d2−1) > d2−2+α (Lemma 2.8). Even when
the path L1 uses the first row of C12 by (∗∗), there is no loss of generality by assuming that
the full rows of C̄12 are among the first α + 1 rows of C̄12. It follows that every row of
Aα+1 has a free entry.
Aα+1
 α + 1 rows d1 – α rows︸ ︷︷ ︸
C12
s2
t2
t4t1
s1
s4
t3
︸ ︷︷ ︸
C̄12
s3
s2
t′2
t′4t1
s1
s′4
t′3
s3
s′2
s′3
L1 s2
t̄2
t̄4t1
s1
s′4
t̄3
s3
s′2
s′3
L1
s̄2
s̄4
s̄3
(a) (b) (c)
Figure 3: Auxiliary figure for Case 2 (a) This shows a scenario where d1 = 5, d2 = 3, and
α = 2. (b) The path L1 = s1−t1 in dashed line, the paths that send the terminals in B1\B3
other than s1 and t1 to B3, and the resulting new linkage Y ′ = {{s′2, t′2}, {s′3, t′3}, {s′4, t′4}}
in C̄12 ∪ Bα+1. (c) The paths that send the terminals in B3 to A3, and the resulting new
linkage Y ′′ = {{s̄2, t̄2}, {s̄3, t̄3}, {s̄4, t̄4}} in C̄12.
Next we show how to send to Bα+1 the terminals other than s1 and t1 that are in the
rows 2 to α + 1 of C12; that is, the terminals other than s1 and t1 that are in B1 \ Bα+1.
For α = 0, B1 \ Bα+1 = ∅ and there is nothing to do. We now focus on the subcase
2 ≤ α ≤ d1 − 2. Let n1 and n2 denote the number of terminals in B1 \ Bα+1 and Bα+1,
respectively. Then the following inequalities hold
n1 + n2 ≤ d1 + 2− α ≤ d1 (since 2 ≤ α),
n1 + n2 ≤ d1 + 2− α ≤ 2d1 − 2α = |V (Bα+1)| (since α ≤ d1 − 2).
From the second inequality, it follows that there are at least n1 free vertices in Bα+1. Since
B1 is d1-connected by Lemma 2.5, Menger’s theorem gives n1 disjoint paths in B1 from the
terminals in B1 \Bα+1 to n1 free entries in Bα+1, avoiding the n2 terminals in Bα+1. For
a terminal si in B1 \Bα+1, let L′i be the path from si to Bα+1 and let s′i := V (L′i)∩Bα+1.
Define t′i similarly for a terminal ti in B1 \ Bα+1. Furthermore, for si (respectively, ti) in
L. K. Jørgensen et al.: Linkedness of Cartesian products of complete graphs 325
Bα+1 ∪ C̄12, let s′i := si (respectively, t′i := ti). This produces a new linkage problem
Y ′ := {{s′2, t′2}, . . . , {s′k, t′k}} in C̄12 ∪Bα+1. See Figure 3(b).
There are at most d1 + 2 − α terminals in Bα+1 = Kd1−α × K2, and every row
in Aα+1 = Kd1−α × Kd2−1 has a free entry. Hence, Claim 2.10 applies, and there
is a linkage formed by X-valid paths from the terminals in Bα+1, other than s1 and
t1, to free entries in Aα+1. For every such path L′′i , if s
′
i ∈ V (L′′i ) ∩ V (Bα+1), let
{s̄i} := V (L′′i )∩V (Aα+1), and if t′i ∈ V (L′′i )∩V (Bα+1), let {t̄i} := V (L′′i )∩V (Aα+1).
Besides, for s′i ∈ C̄12 (respectively t′i ∈ C̄12), let s̄i = s′i (respectively, t̄i = t′i). This pro-
duces a new linkage problem Y ′′ := {{s̄2, t̄2}, . . . , {s̄k, t̄k}} in C̄12 whose feasibility
implies that of Y ′, and therefore that of Y in G, by completing each linkage problem with
the path L1. See Figure 3(c).
Now we have a new linkage problem Y ′′ in C̄12 with (k− 1) pairs. The solution of Y ′′
in C̄12 implies a solution of the linkage problem Y in G. To link the pairs of Y ′′ use the
(k − 1)-linkedness of C̄12.
Finally assume that α = 1. Then there are exactly d1 + 1 terminals in C12 and at most
d2 − 1 terminals in C̄12. In a first scenario suppose that either both entries in B1 \ B2 are
nonterminals or each terminal other than s1 and t1 in B1 \B2 is adjacent to a nonterminal
in B2. Then we can send these terminals in B1 \B2 to B2. In the second scenario, suppose
that there is a terminal si (i ̸= 1) in B1 \ B2 whose neighbours in B2 are all terminals.
Then the column of si in B1 would contain exactly d1 terminals, including si. We send si
to a free entry in A1, in the same row as si (the first row of A1): if this free entry didn’t
exist, then si would be adjacent to the d2 − 1 terminals in A1 and the d1 − 1 terminals in
B2. Since there are d1 + d2 terminals in total, it would follow that si is adjacent to ti. This
contradiction shows that we can send si to a free entry in A1.
In both scenarios, it remains to send the terminals other than s1 and t1 in B2 = Kd1−1×
K2 to A2 = Kd1−1 ×Kd2−1. To do so, we reason as in the subcase 2 ≤ α ≤ d1 − 2. It
follows that there are at most d1 + 2 − 1 terminals in B2, and that every row in A2 has a
free entry. Claim 2.10 applies again and gives a linkage formed by X-valid paths from the
terminals in B2, other than s1, t1, to free entries in A2.
With all the terminals other than s1 and t1 in C̄12, therein we have a new linkage
problem Y ′ with k−1 pairs whose solution in C̄12 implies a solution of the linkage problem
Y in G. To solve Y ′ in C̄12 use the (k − 1)-linkedness of C̄12.
By symmetry, we also have the result if there are at most d2 + 2 terminals in R12,
including {s1, t1}.
Case 1. The subgraph C12 contains at least d1 + 3 terminals, including {s1, t1}.
This case reduces to the previous case. If C12 contains at least d1 + 3 terminals then
R12 contains at most d2 − 3 + 4 = d2 + 1 terminals, since there are four entries shared
by C12 and R12. Because we make no distinction between columns and rows, this case is
already covered. This completes the proof of the theorem.
3 Duals of cyclic polytopes
There is a close connection between duals of cyclic d-polytopes with d + 2 vertices and
Cartesian products of complete graphs.
The moment curve in Rd is defined by x(t) := (t, t2, . . . , td) for t ∈ R, and the convex
hull of any n > d points on it gives a cyclic polytope C(n, d). The combinatorics of a cyclic
326 Ars Math. Contemp. 22 (2022) #P2.09 / 317–326
polytope, the face lattice of the polytope faces partially ordered by inclusion, is independent
of the points chosen on the moment curve. Hence we talk of the cyclic d-polytope on n
vertices [6, Example 0.6].
For a polytope P that contains the origin in its interior, the dual polytope P ∗ is defined
as
P ∗ = {y ∈ Rd | x · y ≤ 1 for all x in P}.
If P does not contain the origin, we translate the polytope so that it does. Translating the
polytope P changes the geometry of P ∗ but not its face lattice. The face lattice of P ∗ is
the inclusion reversed face lattice of P . In particular, the vertices of P ∗ correspond to the
facets of P , and the edges of P ∗ correspond to the (d− 2)-faces of P . The dual graph of a
polytope P is the graph of the dual polytope, or equivalently, the graph on the set of facets
of P where two facets are adjacent in the dual graph if they share a (d− 2)-face.
Duals of cyclic d-polytopes are simple d-polytopes. It is also the case that the dual
of a cyclic d-polytope with d + 2 vertices can be expressed as T (⌊d/2⌋) × T (⌈d/2⌉)
([6, Example 0.6]). From this observation and Theorem 2.1 the next corollary follows
at once.
Corollary 3.1. Duals of cyclic polytopes with d + 2 vertices are ⌊d/2⌋-linked for every
d ≥ 2.
ORCID iDs
Leif K. Jørgensen https://orcid.org/0000-0003-4922-3937
Guillermo Pineda-Villavicencio https://orcid.org/0000-0002-2904-6657
Julien Ugon https://orcid.org/0000-0001-5290-8051
References
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[3] J. E. Goodman and J. O’Rourke (eds.), Handbook of Discrete and Computational Geometry,
Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 1997,
http://www.csun.edu/˜ctoth/Handbook/HDCG3.html.
[4] G. Mészáros, On linkedness in the Cartesian product of graphs, Period. Math. Hungar. 72 (2016),
130–138, doi:10.1007/s10998-016-0113-8.
[5] S. Špacapan, Connectivity of Cartesian products of graphs, Appl. Math. Lett. 21 (2008), 682–
685, doi:10.1016/j.aml.2007.06.010.
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Verlag, New York, 1995, doi:10.1007/978-1-4613-8431-1.
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.10 / 327–361
https://doi.org/10.26493/1855-3974.2433.ba6
(Also available at http://amc-journal.eu)
Characterization of a family of rotationally
symmetric spherical quadrangulations
Lowell Abrams
The George Washington University, Washington DC 20052, USA
Daniel Slilaty *
Wright State University, Dayton OH 45435, USA
Received 11 September 2020, accepted 29 August 2021, published online 27 May 2022
Abstract
A spherical quadrangulation is an embedding of a graph G in the sphere in which each
facial boundary walk has length four. Vertices that are not of degree four in G are called
curvature vertices. In this paper we classify all spherical quadrangulations with n-fold
rotational symmetry (n ≥ 3) that have minimum degree 3 and the least possible number
of curvature vertices, and describe all such spherical quadrangulations in terms of nets of
quadrilaterals. The description reveals that such rotationally symmetric quadrangulations
necessarily also have a pole-exchanging symmetry.
Keywords: Quadrangulation, spherical quadrangulation, rotational symmetry.
Math. Subj. Class. (2020): 05C10
1 Introduction
If S is a closed surface, a graph G embedded in S in which all facial boundary walks
have length four is called a quadrangulation of S. When S is the sphere, the graph G
is necessarily bipartite. Considering quadrilateral faces to be geometrically flat squares,
vertices of degree 4 extend this flatness to neighboring faces. Thus, when “most” of the
vertices of a spherical quadrangulation are of degree four, large areas will appear as a
portion of the geometrically-flat, infinite {4, 4}-planar lattice. The curvature is therefore
localized at vertices of degree other than 4.
In spherical triangulations where each face is considered to be a flat equilateral triangle,
vertices of degree 6 play a similar role in extending flatness, and curvature is thus localized
*Corresponding author.
E-mail addresses: labrams@gwu.edu (Lowell Abrams), daniel.slilaty@wright.edu (Daniel Slilaty)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
328 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
Figure 1: A flat region with all internal vertices of degree 4.
at vertices of degree other than 6. Such spherical triangulations link to several well-studied
structures; for example, those for which all of the curvature is localized at 12 vertices of
degree 5 were popularized by the geodesic domes of Buckminster Fuller. Moreover, these
triangulations were later noted to be the topological dual graphs of what are now called
fullerene graphs. ([4] is a good starting source for fullerenes and other chemical graphs.)
As such, in a quadrangulation of S, a vertex of degree other than 4 is called a curva-
ture vertex. In this paper we investigate spherical quadrangulations with n-fold rotational
symmetry (n ≥ 3) that have minimum degree 3 and the least possible number of curva-
ture vertices, which is 8 when n = 4 and 2n + 2 otherwise. (See Proposition 2.1 and
Corollary 2.2.) The fact that curvature is localized at a relatively small number of vertices
suggests that G may have a description in terms of a geometric net of polygons. For ex-
ample, in Figure 2 we have a net of six congruent quadrilaterals which closes up to yield a
spherical quadrangulation with 3-fold rotational symmetry about poles p and q. Note that
there are 2n+ 2 = 8 curvature vertices of degree 3 each.
q
q
q
p
p
p
Figure 2: A quadrilateral net which describes a spherical quadrangulation with 3-fold rota-
tional symmetry.
Our main result (Theorem 1.1) is that every such spherical quadrangulation has a similar
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 329
description as a very simple net of 2n congruent quadrilaterals. The Three- and Four-
Parameter Constructions mentioned in Theorem 1.1 are stated in full detail in Section 3.
We also identify which of these nets produce quadrangulations that are overlay graphs of
self-dual planar embeddings.
Theorem 1.1. Let G be a spherical quadrangulation with minimum degree 3, having n-fold
rotational symmetry (n ≥ 3), and having the least possible number of curvature vertices.
If
(1) all white vertices have degree 4 and
(2) the poles of the rotational symmetry are at two black vertices,
then G can be obtained from either the Three- or Four-Parameter Construction of Section 3.
Remark 1.2. The reader may note that not all spherical quadrangulations with n-fold rota-
tional symmetry and the least possible number of curvature vertices satisfy Conditions (1)
and (2) of Theorem 1.1. Nevertheless, any such spherical quadrangulation can still be ob-
tained using Theorem 1.1 by making use of Proposition 1.3. In short, even if G does not
satisfy both (1) and (2), it is necessarily the case that G overlayed with its topological dual
graph will be a spherical quadrangulation which does satisfy both (1) and (2). This will be
described in more detail in Section 1.1 in the paragraph after Proposition 1.3.
The reader may note that the quadrangulation of Figure 2, and indeed all of the quad-
rangulations constructed in Section 3, not only possesses n-fold rotational symmetry at
poles p and q but also has a 2-fold symmetry which exchanges p and q. This is interesting
in that this 2-fold symmetry is not a priori implied by our hypotheses. Hence our spher-
ical quadrangulations possess, at the very least, an order-2n symmetry group. Any such
quadrangulations possessing additional symmetries will, of course, also be included in our
constructions.
1.1 Overlay graphs and other background on quadrangulations
An important fact about quadrangulations is that a spherical quadrangulation is always
bipartite while quadrangulations of other surfaces need not be. Given any graph H that is
cellularly embedded in a closed surface S, two bipartite quadrangulations that are naturally
associated with the embedding of H and its topological dual graph H∗ are the overlay
graph and the radial graph. For the sphere, in fact, any quadrangulation G is a radial graph
for some embedding H and its dual H∗. Applications of radial graphs, overlay graphs, and
the closely associated medial graph can be found in [1, 2, 3, 5, 7, 8, 9].
Consider a graph H cellularly embedded in a closed surface S and also consider its
topological dual graph H∗. Say that all of H (both vertices and edges) is colored “red” and
all of H∗ is colored “blue”. Embed H and H∗ simultaneously in S and at each edge/dual-
edge crossing point create a new vertex of degree four (which now has alternating red and
blue edges in rotation around it) and say that this new vertex is “white”. The graph obtained
is called the overlay graph O(H,H∗). Note that H is self dual if and only if O(H,H∗)
has a cellular automorphism which leaves white invariant and switches red and blue colors.
The overlay graph O(H,H∗) was used by Servatius and Servatius [8] to classify self-
dual embeddings in the sphere along with the pairing of their groups of color-preserving
cellular automorphisms of O(H,H∗) as index-2 subgroups of the groups of the white-
preserving cellular automorphisms. Graver and Hartung [5] do the same in the special case
330 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
of self-dual embeddings of graphs having four trivalent vertices and the remaining vertices
all of degree four. Their results, however, are much more explicit than those in [8].
The overlay graph O(H,H∗) is a bipartite quadrangulation of a closed surface S with
partite sets Red ∪ Blue and White. The radial graph of H and H∗ (denote it by R(H,H∗))
can be constructed from O(H,H∗) by placing a diagonal edge in each face of O(H,H∗)
which connects the red and blue vertices on that face and then erasing all of the edges
and white vertices of O(H,H∗). Thus R(H,H∗) is also a bipartite quadrangulation of
S. Conversely, if G is a bipartite quadrangulation of S with bipartition Red ∪ Blue, then
G = R(H,H∗) for some H and H∗ as follows. In each face of G, place a red edge
connecting the two red vertices and a blue edge connecting the two blue vertices. The
resulting red and blue graphs are H and H∗.
Unlike the radial graph, even if G is a bipartite quadrangulation of S in which all white
vertices have degree four it is not necessarily true that G is of the form O(H,H∗) for some
H . An additional condition that does ensure that G has the form O(H,H∗) is given in
Proposition 1.3. In Proposition 1.3, D(G) is the graph obtained from quadrangulation G
by placing a diagonal edge connecting the black corners of each face and then deleting the
white vertices of G. As mentioned in the previous paragraph, D(G) is the radial graph for
some K and K∗ in S when D(G) is bipartite.
Proposition 1.3. If G is a quadrangulation of closed surface S, then G = O(H,H∗) for
some H if and only if every white vertex of G has degree 4 and both G and D(G) are
bipartite. In the particular case that S is the sphere, G = O(H,H∗) for some H if and
only if every white vertex of G has degree 4.
Proof. First, let S be any closed surface. The one direction of the equivalence statement is
trivial. For the other direction, the fact that D(G) is bipartite allows us to properly 2-color
(red and blue) the vertices of D(G), which shows G is of the form O(H,H∗), as required.
The statement for the sphere follows from the first statement and the fact that any spher-
ical quadrangulation is automatically bipartite.
Now say H is a spherical quadrangulation with n-fold rotational symmetry and the
minimum number of curvature vertices, but does not satisfy the other two conditions in
Theorem 1.1. In this case O(H,H∗) inherits the n-fold rotational symmetry of H and
does satisfy the two additional conditions.
Quadrangulations have also been studied for other surfaces. Thomassen [10] and also
Márquez, de Mier, Noy, Revuelta [6] give explicit constructions for all 4-regular quadran-
gulations of the torus and Klein bottle. If G is a quadrangulation of S having no curvature
vertices, then in fact S must be the torus or Klein bottle. (See Proposition 2.1.)
2 Basic properties of spherical quadrangulations
Proposition 2.1 gives an arithmetic constraint on the quantities and degrees of curvature
vertices in quadrangulations, and Corollary 2.2 is an immediate consequence. We use χ(S)
to denote the Euler Characteristic of the surface S.
Proposition 2.1. If G is a quadrangulation of closed surface S with minimum degree 3
and vi vertices of degree i then,
v3 = 4χ(S) +
∑
i≥5
(i− 4)vi.
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 331
Furthermore, if χ(S) ̸= 0, then there are curvature vertices.
Proof. If G has f faces and e edges, then
∑
i ivi = 2e. Also, 4f = 2e and
(∑
i vi
)
− e+
f = χ(S) which when combined together yield 4(
∑
i vi) = 4χ(S) + 2e. Now subtracting
we obtain
∑
i(4− i)vi = 4χ(S) which yields our desired results.
Corollary 2.2. If G is a spherical quadrangulation with n-fold rotational symmetry
(n ≥ 3), minimum degree 3, and having the least possible number of curvature vertices,
then
• if n = 3, then G has eight vertices of degree 3 of which two are poles of the rotational
symmetry;
• if n = 4, then G has 8 vertices of degree 3 and the two poles of the rotational
symmetry are either both at vertices of degree 4 or both at the centers of faces; and
• if n > 4, then G has two vertices of degree n and 2n vertices of degree 3 where the
two vertices of degree n are the two poles of the rotational symmetry.
In [5], Graver and Hartung give a complete construction of spherical quadrangulations
of the form G = O(H,H∗) where H ∼= H∗ is a planar graph with four vertices of degree
3 and all other vertices of degree 4. (They do not assume any rotational symmetry). Here,
for n = 3, we assume only that G is a spherical quadrangulation with 3-fold rotational
symmetry and discover structures not found in [5].
Given a vertex v in a graph G and a subgraph H of G, the difference dG(v) − dH(v)
(that is, the degree of v in G minus the degree of v in H) is called the codegree of v with
respect to H and G. We say that v is saturated by a subgraph H when v has codegree zero
with respect to H and G.
Consider a spherical quadrangulation G with topological dual graph G∗. A collection
X of faces of G corresponds to a collection of vertices X∗ of G∗. If the induced subgraph
of G∗ on vertex set X∗ is connected, then the union of the faces in X along their incident
edges and vertices is called a face-connected subsurface. Let F be the face-connected
subsurface corresponding to X . The boundary of F , call it ∂F , is the collection of edges
(and their endpoints) incident to exactly one face in X . Topologically speaking, a face-
connected subsurface F is a sphere with holes (including the possibility of no holes) where,
of course, if there is exactly one hole, then F is topologically a disk. The boundary ∂F
is now an edge-disjoint union of cycles in G bounding the holes of F . The total length of
the union of boundary cycles is called the circumference or total circumference of F . An
interior vertex of F is a vertex not on ∂F while a boundary vertex is a vertex on ∂F .
The distance between two vertices u and v in a graph G is the length (edge length) of a
shortest uv-path in G. Denote this distance by dG(u, v). Of course, dG(u, v) is even iff u
and v are either both black or both white. Given a vertex v in a spherical quadrangulation G,
consider a face F with white vertices w1 and w2 and black vertices b1 and b2. For any vertex
v in G, evidently, |dG(v, wi)− dG(v, bj)| = 1 for each i and j ∈ {1, 2}. Additionally, the
following three possibilities may occur for F with respect to v: dG(v, w1) = dG(v, w2)
and |dG(v, b1)− dG(v, b2)| = 2; |dG(v, w1)− dG(v, w2)| = 2 and dG(v, b1) = dG(v, b2);
and dG(v, w1) = dG(v, w2) and dG(v, b1) = dG(v, b2).
Given a vertex v and a face f in a spherical quadrangulation G, let u be a vertex on f
of smallest distance from v, say distance t. The vertices in cyclic ordering around f now
332 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
have distances t, t + 1, t + d, t + 1 from v where d ∈ {0, 2}. Given integer k ≥ 0, define
Xk(v) to be the set of faces f for which t + 1 ≤ k. The face-connected subsurface of G
given by Xk(v) in Proposition 2.3 is called the k-ball centered at v and is denoted Bk(v).
Proposition 2.3. The faces in Xk(v) define a face-connected subsurface of G.
Proof. Certainly X1(v) is a face-connected subsurface. Using induction, assume that
Xk−1(v) is a face-connected subsurface and consider a face f in Xk(v) that is not in
Xk−1(v). We will complete the proof by showing that there is a facial path (that is, a path
in G∗) from f to a face in Xk−1(v). Let u be a vertex on f whose distance from v is the
smallest and let x1 and x2 be the neighbors of u on f . If we write dG(v, u) = t, then
dG(v, x1) = dG(v, x2) = t+ 1. It must be that t+ 1 = k or else f would be in Xk−1(v).
Now consider the neighbors of u in G, say x1, x2, . . . , xm. Since dG(v, u) = k − 1,
we get that dG(v, xi) ∈ {k−2, k} for each i which implies that all faces of G incident to u
are in Xk(v). Now there must be some i for which dG(v, xi) = k− 2 and so there is some
face f ′ incident to u which is in Xk−1(v). The rotation of faces in G around u contains a
path of adjacent faces from f to f ′.
The reader can easily verify that Proposition 2.4 follows directly from definitions.
Proposition 2.4. If v is a vertex in a spherical quadrangulation G and k ≥ 1, then the
following hold.
(1) If e is an edge of G whose vertices have distances t and t+1 from v with t+1 ≤ k,
then both faces of G incident to e are in Bk(v).
(2) If u is a vertex of G having distance t ≤ k from v, then u is in Bk(v).
(3) If f is a face of Bk(v) sharing an edge with ∂Bk(v), then the vertices of f have
distances k−1, k, k+1, k from v; furthermore, an edge of f on ∂Bk(v) has endpoints
with distances k and k + 1 from v.
(4) If f ′ is a face that is not in Bk(v) and which shares an edge with ∂Bk(v), then the
vertices of f ′ have distances k, k + 1, k + d, k + 1 from v where d ∈ {0, 2}.
(5) The vertices of a cycle C on ∂Bk(v) have distance from v alternating k and k + 1.
k or k + 2
k
k + 1
k + 1
f ′f
k − 1
k
Figure 3: Proposition 2.4(4).
A standard k-disk with n-fold rotational symmetry around a fixed black vertex is con-
structed as follows. Consider the standard {4, 4} planar quadrangulation and designate
one black vertex as an origin, and then label perpendicular x- and y-axes. Consider the
part of the quadrangulation in the first quadrant, with coordinate axes included, consisting
of the union of the closed faces whose interiors are either beneath or intersecting the line
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 333
x+ y = k (see Figure 4). Call this planar graph a k-wedge. Taking n copies of a k-wedge,
W1, . . . ,Wn, identify the x-axis of Wi with the y-axis of Wi+1 (subscript addition taken
modulo n) with origin vertex identified to origin vertex to obtain the standard k-disk. In
Figure 5 we show the standard 3- and 4-disks with a central vertex of degree n = 5.
y
x
Figure 4: The 11-wedge.
Figure 5: Standard 3- and 4-disks.
Consider a degree-4 vertex in G with incident edges e1, e2, e3, e4 in rotational order.
We call each of the pairs e1, e3 and e2, e4 transverse. A path P in a planar graph G is said
to be a transverse if each of its interior vertices has degree four in G with consecutive edges
on P forming a transverse pair. In a standard k-disk, there are n distinct transverse paths
of length k emanating from the origin (call it p) which we call the central rays. Note that
every vertex of distance k from p in a standard k-disk K has degree 4 in K aside from the
n endpoints of the central rays which have degree 3 in K. The vertices of distance k + 1
from p in K have degree 2 in K.
Proposition 2.5. The circumference of the standard k-disk is 2nk and the number of faces
in the standard k-disk is n
(
k+1
2
)
.
334 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
Proof. The number of faces in the first quadrant of the coordinatized {4, 4} planar quad-
rangulation that intersect the line x+ y = k is k. Therefore the number of boundary edges
of the k-wedge is 2k and the number of faces in the k-wedge is a triangular sum. The result
follows.
Proposition 2.6. If G is a spherical quadrangulation and Bk(p) is a standard k-disk in G,
then every vertex of distance k from p is on ∂Bk(p). Furthermore, any vertex of distance
k + 1 from p is either on ∂Bk(p) or is a neighbor of some vertex of distance k from p on
∂Bk(p) that is not saturated by Bk(p).
Proof. Here ∂Bk(p) is a single cycle separating the sphere into regions R1 and R2 where
without loss of generality p is in R1. Since Bk(p) is a standard k-disk, we know that every
interior vertex of Bk(p) has distance less than k from p. Given a vertex v in R2, a shortest
pv-path in G must pass through ∂Bk(p) and so v has distance strictly larger than k. This
implies our desired result.
Proposition 2.7. If D is a disk in a spherical quadrangulation G defined by a set of faces
X with |X| ≥ 2, then there are two distinct faces f1, f2 ∈ X such that each X−fi defines
a disk D′ for which the intersection of the boundary of face fi with ∂D′ is a path.
Proof. Consider a face f ∈ X whose boundary shares an edge with ∂D, call this a bound-
ary face. Using the fact that D is a disk, the reader can confirm that the following are
equivalent. (Here G∗[Y ∗] is the induced subgraph of G∗ on vertices Y ∗ where Y is a
collection of faces in G.)
• The faces X − f define a disk.
• The faces X − f define a face-connected subsurface of G, that is, G∗[X∗ − f∗] is
connected.
• The intersection of f with ∂D is a single path.
Assume by way of contradiction that G∗[X∗ − f∗] is disconnected for each boundary
face f ∈ X . By disconnectedness, the degree of f∗ in G∗[X∗] is not 1; furthermore, since
f is a boundary face, the degree of f∗ in G∗[X∗] is either 2 or 3. The number of connected
components of G∗[X∗ − f∗] is two when the degree is 2 and is two or three when the
degree is 3. We get 3 connected components precisely when f intersects ∂D in three paths
of lengths 1, 0, and 0. Each connected component of G∗[X∗ − f∗] must contain a vertex
corresponding to a boundary face of D.
Let f ∈ X be a boundary face for which the induced subgraph of G∗[X∗−f∗] contains
a connected component on vertex set C ⊆ X∗ − f∗ with |C| as small as possible. Pick
f∗0 ∈ C that is a boundary face of D. By assumption, G∗[X∗ − f∗0 ] is disconnected;
however, by planarity, one of its connected components has vertex set which is a proper
subset of C, a contradiction of minimality.
Given a face f1 ∈ X such that G∗[X∗ − f∗1 ] is connected, we will now find a face
f2 ̸= f1 such that G∗[X∗− f∗2 ] is connected. Since |X| ≥ 2, there must be boundary faces
in D other than f1. By way of contradiction, assume that for every boundary face f2 ̸= f1
we have that G∗[X∗−f∗2 ] is disconnected. Pick f2 such that G∗[X∗−f∗2 ] has a connected
component on vertices C with f∗1 /∈ C and |C| as small as possible. Following the same
argument as above, we will contradict the minimality of C.
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 335
Proposition 2.8 is surely an expected outcome; however, there are subtleties that require
verification. Proposition 2.8 provides a standard model for iteratively building up disks in
G.
Proposition 2.8. Let D be a disk in a spherical quadrangulation G such that D contains
a black vertex p having degree n ≥ 3 in G and all other vertices of D have degree 4 in G.
Then for k large enough, D is isomorphic to a face-connected subsurface of the standard
k-disk with n-fold rotational symmetry around a black central vertex p0. Furthermore, p is
identified with p0.
Proof. Let X be the collection of faces defining D. If |X| = 1, then the result is clearly
true. If |X| ≥ 2, then by Proposition 2.7, there is an ordering f1, . . . , fm of the elements of
X , such that p is on f1 and Xi = {f1, . . . , fi} defines a disk Di such that fi+1 intersects
∂Di in a path. Inductively Di is a face-connected subsurface of a standard k-disk, Sk, for
some large enough value of k. If ∂Di intersects ∂Sk, then increase k by 2 so that ∂Di no
longer intersects the boundary of the k-disk.
Let Pi be the path of intersection of fi+1 with ∂Di (the length of Pi being 1, 2, or 3).
Every internal vertex of Pi must be saturated by Di as a subgraph of G and each endpoint
of Pi is not saturated. Additionally, every vertex of D other than p has degree 4 in G, so
the codegree of any vertex of Di is the same with respect to being a subgraph of G or Sk.
We now have that every internal vertex of Pi is saturated by Di as a subgraph of Sk and
each endpoint of Pi is not saturated. Thus Pi is incident to a unique face f ′ of Sk that is
not in Di. The face f ′ may now be identified with fi+1 and we have Di+1 as a subgraph
of the standard k-disk Sk.
3 The two constructions
We will define two families of spherical quadrangulations, one defined with three indepen-
dent parameters and the other with four. Each spherical quadrangulation is described as a
net of 2n congruent convex quadrilaterals with vertices on the 2-dimensional integer lattice
in which two sides of the quadrilateral are perpendicular and of the same length. For lack
of a more specific term, we will call such a quadrilateral a special integer quadrilateral.
3.1 Three Parameters
Choose positive even integer a, non-negative integer s, and l ∈ {0, . . . , a − 1}. Consider
the special integer quadrilateral of Figure 6. By reflecting along the line y = x we may
assume that l ∈ {0, . . . , a2}
We assemble a net of 2n such quadrilaterals as indicated in Figure 7 to obtain a spherical
quadrangulation with n-fold rotational symmetry with poles at black vertices p and q and
with all white vertices of degree 4. Note that the arrangement of the quadrilaterals in
this construction will always yield a quadrangulation with an order-2 rotational symmetry
which exchanges p and q.
In Proposition 3.2 we characterize when the Three-Parameter Construction yields a
spherical grid which is the overlay graph of a self-dual embedding. Proposition 3.1 gives
us a necessary and sufficient condition for making this characterization.
336 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
(0, 0)
(0, a)
(a, 0)
(s+ l, a+ s− l)
Figure 6: The Three-Parameter Construction with (a, s, l) = (8, 3, 1).
p
p
p
q
q
q
Figure 7: A net constructed from 6 copies of the quadrilateral in Figure 6.
Proposition 3.1. If O(H,H∗) has n-fold rotational symmetry with poles at black vertices,
the minimum possible number of curvature vertices, and an additional symmetry that ex-
changes the poles, then H ∼= H∗ if and only if one pole is in H and the other in H∗.
Proof. Let p and q be the poles of the n-fold rotational symmetry of O(H,H∗). If one of
p and q is in H and the other in H∗, then because R(H,H∗) is bipartite, the symmetry
which exchanges poles must exchange all of H and H∗ and so is an isomorphism between
H and H∗.
Conversely, assume that H ∼= H∗ and say that H is red and H∗ is blue. As such, the
rotational symmetry, which fixes p and q, preserves the red and blue colors whereas the
symmetry which exchanges p and q exchanges red and blue colors.
If n = 3, then four of the degree-3 vertices are red and four are blue and p and q
both have degree 3. Furthermore, the six degree-3 vertices other than p and q are therefore
divided into two orbits under the rotational symmetry of three vertices each, one being red
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 337
and the other blue. Therefore the fourth red degree-3 vertex is one of p and q and the fourth
blue degree-3 vertex is the other.
If n ≥ 4, then the rotational symmetry divides the 2n degree-3 vertices into two orbits
of n vertices each; one must be red and the other blue. Call these Or and Ob. Without loss
of generality, say that p is red. Thus the distance in R(H,H∗) from p to the vertices in Or
is even while the distance to the vertices in Ob is odd. Now the symmetry which exchanges
p and q must therefore exchange Or and Ob, more generally, exchange red and blue colors.
Thus p ∈ H and q ∈ H∗.
Proposition 3.2. A spherical quadrangulation constructed from the Three-Parameter Con-
struction is the overlay graph of a self-dual graph if and only if s and l have different
parities.
Proof. Let G be a quadrangulation constructed using the Three-Parameter Construction
and let p and q be the poles of the rotational symmetry. By Proposition 1.3, G ∼= O(H,H∗)
for some H . By Proposition 3.1, a necessary and sufficient condition for H ∼= H∗ would
without loss of generality be that p ∈ H and q ∈ H∗. This is true if and only if the distance
from p to q in D(G) is odd.
Consider one quadrilateral of the construction. In this quadrilateral, there is a path in
the graph D(G), from (0, 0) to (0, a) of length a. From (0, a) to (s + l, a + s − l), there
is a path in D(G) of length (s + l) + (s − l) = 2s when s ≡ l mod 2 and of length
(s + l − 1) + (s − l − 1) + 1 = 2s − 1 when s ̸≡ l mod 2. Thus there is a path from p
to q in D(G) of length 2a+ 2s when s ≡ l mod 2 and of length 2a+ 2s− 1 when s ̸≡ l
mod 2, as required.
3.2 Four parameters
Choose positive integers a and b of the same parity. Assume that a ≥ b. Choose non-
negative integers h and w of the same parity, not both zero, and which satisfy
− h
w
≤ b− a
b+ a
and − a
b
(a− w) ≤ b+ h.
(In the case that w = 0 say that − hw = −∞.) Consider the special integer quadrilateral of
Figure 8.
(0, 0)
(a, b)
(−b, a)
(a− w, b+ h)
Figure 8: A special integer quadrilateral with (a, b, h, w) = (7, 3, 2, 2).
Given a choice of a and b, the constraints placed on non-negative integers h and w guarantee
that the quadrilateral defined will indeed be convex. The first inequality guarantees that the
338 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
point (a−w, b+h) lies above the line containing (−b, a) and (a, b). The second inequality
guarantees that at x = a−w, the y-coordinate b+ h is greater than the y-coordinate of the
line of slope − ba . (See Figure 9.)
(0, 0)
(a, b)
(−b, a)
(a− w, b+ h)
Figure 9: Constraints on (a, b, h, w) guarantee convexity.
We assemble a net of 2n such quadrilaterals as indicated in Figure 10 to obtain a spher-
ical quadrangulation with n-fold rotational symmetry with poles at black vertices p and q
and with every white vertex of degree 4. As with the three-parameter construction, the four-
parameter construction always yields a quadrangulation with an order-2 rotational symme-
try which exchanges p and q.
q
q
q
p
p
p
Figure 10: A net constructed from 6 copies of the quadrilateral in Figure 8.
Proposition 3.3. If G is constructed from the four-parameter construction with parameters
(a, b, h, w), then G is the overlay graph of a self-dual graph if and only if h and w are both
odd.
Proof. Say G is constructed using the Four-Parameter Construction and let p and q be the
poles of the rotational symmetry. As in the proof of Proposition 3.2, G ∼= O(H,H∗) for
some H (say H is red and H∗ is blue) and a necessary and sufficient condition for H ∼= H∗
would be that the distance from p to q in D(G) is odd.
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 339
Consider one quadrilateral of the construction with p at (0, 0). There is a path in D(G)
from p to the vertex at (a, b) of length a+b when a and b are both even and a+b−1 when a
and b are both odd. Call this length x. There is a path in D(G) from (a, b) to (a−w, b+h)
of length w + h when w and h are both even and of length w + h − 1 when h and w are
both odd. Call this length y. Thus there is a pq-path in D(G) of length 2x + y which is
even when h and w are both even and is odd when h and w are both odd, as required.
4 The two constructions are sufficient
In this section we prove Theorem 1.1. So, throughout this section, let n ≥ 3 be a fixed in-
teger and G a spherical quadrangulation satisfying the hypothesis of Theorem 1.1. Recall
that making use of Proposition 1.3 allows this theorem to cover all spherical quadrangu-
lations with minimum degree three and the minimum number of curvature vertices. Our
first step is to prove Proposition 4.2, which separates the remainder of this proof into two
distinct cases. In Section 4.3 we find that all graphs in the first case are given by the Three-
Parameter Construction. In Section 4.4 we find that all graphs in the second case are given
by the Four-Parameter Construction.
4.1 Initial k-balls are standard disks
Proposition 4.1. If every vertex of Bk(p) aside from p has degree 4 in G, then Bk+1(p) is
a standard (k + 1)-disk.
Proof of Proposition 4.1. First we prove that B1(p) is a standard 1-disk. The n white
neighbors of p are all distinct because G is simple. Now the only way in which B1(p)
is not a disk would be if the black vertices along the boundary walk are not all distinct. If,
by way of contradiction, we assume that these black vertices are not all distinct, then rota-
tional symmetry implies that there is only one black vertex on ∂B1(p). This black vertex
must therefore have degree 2n, a contradiction.
Inductively assume that the result holds up to some k − 1 ≥ 0 in G. Assume, by way
of contradiction, that every vertex v ̸= p of Bk(p) has degree 4 in G and yet Bk+1(p) is not
a standard (k + 1)-disk. Since every vertex v ̸= p of Bk(p) has degree 4 in G, we get that
every vertex v ̸= p of Bk−1(p) has degree 4 in G and so inductively Bk(p) is a standard
k-disk.
By Proposition 2.4, every face f that is not in Bk(p) but shares an edge e with ∂Bk(p)
is in Bk+1(p). Each vertex of distance k from p is in Bk(p) (again by Proposition 2.4)
and so has degree 4 in G. Therefore each edge e on ∂Bk(p) satisfies the following: if e is
incident to a central ray then the two edges of f incident to e are not on ∂Bk(p) and if e
is not incident to a central ray, then there are two consecutive edges of f on ∂Bk(p) (see
Figure 11).
The former type of face we will call a “radial” face and the latter a “notch” face. For a
notch face there cannot be 3 edges of f that are on ∂Bk(v) because this would either force
a vertex on ∂Bk(p) to have degree less that 3 in G, a contradiction, or force two vertices
on ∂Bk(p) to be identified, a contradiction because Bk(p) is a standard k-disk. In Case 1
assume that there is a radial face f in Bk+1(p) having opposing edges that are both on
∂Bk(p). In Case 2, every radial face has exactly one of its edges on ∂Bk(p).
340 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
k
k
k
k
f
k f
e′
Figure 11: A radial face and a notch face.
Case 1: Let e and e′ be opposing edges of radial face f that are both on ∂Bk(p). Since e
and e′ are nonconsecutive on f , they must also be non-consecutive along ∂Bk(p) (see the
left in Figure 11). Since e and e′ are the only edges of f on ∂Bk(p), it must also be the
case that e′ is incident to a central ray of Bk(p) as well (which cannot be the same ray).
Let O be the orbit of e under the n-fold rotational symmetry. (Note that |O| = n.) The
black and white coloring of the vertices, along with orientability, forces edges e and e′ to
both point in the same direction along the disk from their central rays, which implies that
e′ ∈ O. Therefore the orbit of f under the n-fold rotational symmetry does not have order
n and so f contains a pseudofixed point in its interior, a contradiction of the fact that p and
q are the only pseudofixed points of the rotational symmetry of the sphere.
Case 2: Let B denote the face-connected subsurface consisting of Bk(p) along with the
faces not in Bk(p) that share an edge with ∂Bk(p). Note that B ⊆ Bk+1(p). Take two
faces f and f ′ of B that are not in Bk(p) which are consecutive with respect to their edges
on ∂Bk(p). Because no vertex v ̸= p of B ⊆ Bk+1(p) has degree other than 4, if one of
f and f ′ is a notch face then f and f ′ share no edges in common, and if f and f ′ are both
radial faces then they must share one edge in common. Thus B is obtained from a standard
(k + 1)-disk, call it K, after perhaps making identifications along ∂K. (See Figure 12.)
Bk(p) B Bk+1(p)
K
Figure 12: The relationships between B, K, Bk(p) and Bk+1(p).
Note that there can be no facial identifications in obtaining B from K because that forces
the identified face to have two opposite edges on ∂Bk(p) which would put us back into
Case 1. In Case 2.1 there are no identifications, in Case 2.2 there is an edge identification,
and in Case 2.3 there are no edge identifications but there are vertex identifications.
Case 2.1: Here we have that B = K is a standard (k + 1)-disk. In this case, the vertices
on the cycle ∂B in G must have distances from p in G alternating between k+1 and k+2.
If Bk+1(p) = B, then we are done. If not, then there is a face f in Bk+1(p) that is not in
B and shares an edge with ∂B. Because f is in Bk+1(p), the distances of the vertices on f
from p must be k+1, k+2, k+1, k; however, ∂B separates p from f in the sphere which
means that no vertex on f can have distance less than k + 1 from p.
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 341
Case 2.2: Let e and e′ be two edges on ∂K that are identified in obtaining B from K. If e
and e′ are consecutive along ∂K with e having endpoints u and v and e′ having endpoints
v and w, then the distances in K from p to u, v, and w are either k + 1, k + 2, k + 1 or
k+2, k+1, and k+2. In the first case, the identification of e and e′ would create a vertex
in G of degree 1, a contradiction. In the second case, the identification of e and e′ would
create either a vertex of degree 2 in G (again a contradiction) or a vertex of degree 3 in G
that is in Bk(p), a contradiction of our inductive hypothesis.
If e and e′ are in the same orbit under the rotational symmetry of K, then the interior of
e would contain a pseudofixed point of the rotational symmetry; however, p and q are the
only pseudofixed points in G of the rotational symmetry, a contradiction.
Now, given that e and e′ are not consecutive along ∂K and not in the same orbit,
rotational symmetry yields an orbit of n distinct edges e1, . . . , en identified to an orbit of
n distinct edges e′1, . . . , e
′
n. Planarity of G forces the cyclic ordering of these edges along
∂K to be e1, e′1, . . . , en, e
′
n.
If either e1 or e′1 (say e1) is not incident to a central ray of K, then e has an endpoint v
of degree 4 in K. Thus v has distance k+1 from p in K and say without loss of generality
that v is black. The identification of e1 and e′1 would force v to have degree at least five in
B because the black endpoint of e′1 has degree at least 3 in K. This is impossible unless
the vertex resulting from the identification of e1 and e′1 is the other pole q; however, this
implies that the corresponding endpoints of e2, . . . , en and e′2, . . . , e
′
n are also the pole q.
This is impossible because q has degree n in G and such identifications would force q to
have degree at least 2n, which contradicts the fact that G has maximum degree n.
If both e1 and e′1 are incident to central rays, then these two orbits of edges account for
all of the 2n edges on ∂K that are incident to the central rays. In Figure 13 the edges with
the same numbers are identified and therefore, by the rotational symmetry, the endpoints
of the central rays must be identified to the other pole of the rotational symmetry, call it q.
Making these edge identifications results in a surface K ′ that is topologically a sphere
with n holes; that is, ∂K ′ consists of vertex-disjoint cycles C1, . . . , Cn. As discussed
above, there can be no further edge identifications in going from K ′ to B. If there are
vertex identifications in going from K ′ to B, then each identification is between two white
degree-2 vertices on ∂K ′. These white vertices must be on the same cycle Ci. The reason
for this is as follows. If vertex x on Ci is identified to vertex y on Cj , then let Q be a simple
xy-path in K ′ whose interior avoids ∂K ′ (not necessarily a path in the graph). Now any
cycle on K ′ which avoids its boundary (again, not necessarily a cycle in the graph itself)
and separates Ci from Cj must transversely intersect Q an odd number of times. Thus the
spherical embedding G would have two cycles drawn on it which intersect transversely an
odd number of times, a contradiction.
Now say that two white vertices on Ci (call them x and y) are identified in going from
K ′ to B. Note that the black vertices on Ci all have degree 4 in K ′ and the white vertices
on Ci all have degree 2 in K ′ save for one which has degree 3. Let Pi be the xy-path on
Ci which avoids the degree-3 white vertex. After identifying x and y, facial boundaries
of length four and saturated black vertices forces the identification of the adjacent pair of
white vertices on Pi, and so on. These identifications will eventually result either in a white
vertex being forced to have degree 2 in G (a contradiction) or a face in G being forced to
have length 2 (again a contradiction).
Lastly, assuming there are no further vertex or edge identifications, we have K ′ = B.
Let Di be the disk in G bounded by Ci whose faces are not in B. The black vertices on
342 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
p
q q
q
q
q
1 1
2
2
3
34
4
5
5
Figure 13: Edges with the same numbers are identified which then implies that the vertex
q is the other pole.
Ci have degree 2 in Di and the white vertices on Ci have degree 4 in Di save for one
white vertex which has degree 3 in Di. By rotational symmetry, there must be two black
degree-3 vertices in the interior of Di, with the remaining interior vertices having degree
4. This forces Di to have exactly three vertices of odd degree, a contradiction.
Case 2.3: If two identified vertices on ∂K are in the same orbit under the rotational sym-
metry, then the resulting vertex will be pseudofixed and so the identified vertex is the pole
q. However, q will now be forced to have degree at least 2n, a contradiction. So now
take an orbit of n distinct vertices v1, . . . , vn on ∂K that are pairwise identified to the n
distinct vertices v′1, . . . , v
′
n in going from K to B. There can be no additional identifica-
tions among these vertices. Planarity now forces these 2n vertices to have cyclic ordering
v1, v
′
1, . . . , vn, v
′
n along ∂K. Thus the vertex v1 = v
′
1 in G has degree 4 in G. So now
if K ′ is obtained from K by making these n identifications only, then K ′ is obtained as
shown in Figure 14 (for n = 5).
v1 v′1
v2
v′2
v3
v′3v4
v′4
v5
v′5
Figure 14: The surface K ′ from Case 2.3.
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 343
Rechoose the identified vertex pairs v1, v′1, . . . , vn, v
′
n so that vi and v
′
i are as close together
along ∂K as possible. Thus the orbit of n holes in K ′ have boundaries that are cycles in
G, say C1, . . . , Cn. Now either the endpoint of a central ray of K ′ is on C1 and is adjacent
to v1 = v′1 or not. Let these be Cases 2.3.1 and 2.3.2.
Case 2.3.1: Consider the disk H in G with ∂H = C1 whose faces are not in K. By
rotational symmetry either 0, 1, or 2 of the degree-3 vertices of G appear in H and the pole
q does not appear in H . Assume for the moment that 0 of the degree-3 vertices of G appear
in H . Say that C1 has length 2m. The degrees in H of the vertices on C1 are therefore
2, 3, 4, 2, 4, 2, . . . , 4, 2 in which the first degree-2 vertex is v1 = v′1 and the degree-3 vertex
is the endpoint of the central ray of K ′. The remaining vertices of H all have degree 4.
Thus the sum of the degrees of the vertices in H is
4i+ 3 + 4(m− 1) + 2m = 4i+ 6m− 1
where i is the number of interior vertices. So now if ϵ is the number of degree-3 vertices
of G appearing in H , then the sum of the degrees of the vertices in H is 4i+ 6m− 1− ϵ.
Now if e is the number of edges in H , we obtain
4i+ 6m− 1− ϵ = 2e.
If f is the number of quadrilateral faces in H , then
4f + 2m = 2e.
Now Euler’s Formula implies that
1 = i+ 2m− e+ f = 1
4
(2e+ ϵ+ 1− 6m) + 2m− e+ 1
4
(2e− 2m) = 1
4
(1 + ϵ) ≤ 3
4
,
which is a contradiction.
Case 2.3.2: Let u1 and w1 be the neighbors of v1 = v′1 on C1. Note that these three vertices
all have degree 4 in K ′ and so have no edges extending into the interior of H . This forces
these three vertices to be on the same quadrilateral face of G and this face is in H . Because
v1 and v′1 are chosen to be as close together as possible along ∂K, it must be that v1 and v
′
1
are at distance 4 apart along ∂K. Hence C1 has length four and H has only one face. Let
x be the fourth vertex of C1. Since u1 and w1 both have degree 4 in K ′ and K, it must be
that x has degree 2 in K ′ which implies that x also has degree 2 in G, a contradiction.
Proposition 4.2. If every vertex v ̸= p of Bk−1(p) has degree 4 in G but there are curvature
vertices of G in Bk(p)− p, then Bk(p) is a standard k-disk and the following hold.
(1) If k is even, then the n endpoints of the central rays of Bk(p) have degree 3 in G and
all other vertices of Bk(p)− p have degree 4 in G.
(2) If k is odd, then there are either n or 2n degree-3 vertices of G which have distance
k + 1 from p on ∂Bk(p) and all other vertices of Bk(p) − p have degree 4 in G,
including the endpoints of the central rays.
Proof. By Proposition 4.1, Bk(v) is a standard k-disk. By Proposition 2.4, the vertices of
Bk(p) that are not in Bk−1(p) come in two types: those on ∂Bk(p) having distance k + 1
344 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
from p and the endpoints of the central rays of Bk(p), which have distance k from p. If k is
odd, then because the curvature vertices of G are black, Bk(p) is of Type (2). If k is even,
then because the curvature vertices of G are black, Bk(p) is of Type (1) and there are either
n or 2n such curvature vertices.
Proposition 4.2 gives us two cases for G. In Section 4.3 we will discuss the case for G
in Part (1) of Proposition 4.2 and in Section 4.4 the case for Part (2) of Proposition 4.2.
4.2 Necklaces
Take quadrilaterals q1, . . . , qn whose vertices are properly colored black and white. Label
the black vertices of qi with bi,1 and bi,2. A diamond necklace of length n with a black
diagonal is the graph obtained from q1, . . . , qn by identifying bi,2 with bi+1,1 for each
i ∈ {1, . . . , n} where addition in subscripts is taken modulo n so as to obtain a cyclic
arrangement of these quadrilaterals. The top of Figure 15 shows a diamond necklace with a
black diagonal. A diamond necklace of length n with a white diagonal is defined similarly.
When t diamond necklaces of the same length with diagonals of alternating colors are
stacked together as on the bottom of Figure 15, we obtain a straight thorax of thickness t.
xx
x
y
z
x
y
z
Figure 15: A diamond necklace with a black diagonal and a straight thorax of thickness 5.
Consider a diamond necklace of length n(k + 1) for some k ≥ 1 with black-diagonal
vertices b0, . . . , bn(k+1)−1 and choose a positive integer 1 ≤ l ≤ k. If we embed our
necklace in the plane with b0, . . . , bn(k+1)−1 oriented in the clockwise direction, then there
are well-defined inner and outer boundary cycles of length 2n(k + 1) each. For each
vertex i(k + 1) ∈ {0, k + 1, . . . , (n − 1)(k + 1)} identify the two inner-boundary edges
incident to bi(k+1) and identify the two outer-boundary edges incident to bi(k+1)+l. The
resulting graph with n-fold rotational symmetry is called a (k, l)-zig-zag necklace with a
black diagonal. Note that the lengths of each of the two boundary cycles of the (k, l)-zig-
zag necklace is 2nk. A (k, l)-zig-zag necklace with a white diagonal is defined similarly.
The graph in Figure 16 shows a portion of a (k, 6)-zig-zag necklace.
Any number of (k, l)-zig-zag necklaces with diagonals of alternating colors may be
stacked as shown in Figure 17.
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 345
b0
b1
b2
b3
b4
b5
b6
Figure 16: A portion of a (k, 6)-zig-zag necklace.
Figure 17: Stacking zig-zag necklaces.
4.3 Curvature vertices on the ends of central rays
Proposition 4.3. Let Bk(p) be a standard k-disk in G as described in Proposition 4.2(1). If
the only curvature vertices of Bk+t(p) are p, v1, . . . , vn, then Bk+t+1(p) is a disk obtained
from the standard disk Bk(p) by adding a straight thorax with thickness t+1. Furthermore,
the circumferences of Bk(p) and Bk+t(p) are the same.
Proof. First we observe that the statement about the circumference of Bk+t(p) is evident
by the structure of necklaces. We now proceed with the rest of the proof. Certainly for
t = 0, Bk+t(p) is a disk obtained from the standard disk Bk(p) by adding a straight thorax
with thickness t = 0. So now assume that this same statement holds for some t ≥ 0
and the only curvature vertices of Bk+t(p) are p, v1, . . . , vn. Also as part of the induction
hypothesis we include that ∂Bk+t(p) has vertices in one color class (those of distance k+ t
from p) saturated by Bk+t(p) and vertices in the other color class (those of distance k+t+1
from p) having degree 2 in Bk+t(p).
Now consider Bk+t+1(p). Every face f of Bk+t+1(p) that is not in Bk+t(p) yet shares
an edge with ∂Bk+t(p) must share two consecutive edges with ∂Bk+t(p) because the ver-
tices in one color class of ∂Bk+t(p) are saturated by Bk+t(p). Furthermore, since Bk+t(p)
is a disk, f cannot share three edges with ∂Bk+t(p) because if it did, then f would have
two vertices that are saturated by Bk+t(p) which would force f to share all four of its edges
with ∂Bk+t(p). Since ∂Bk+t(p) is a cycle, this would imply that the length of ∂Bk+t(p) is
four; however, it must have length at least 2n ≥ 6.
346 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
Let B be the face-connected subsurface obtained from Bk+t(p) by adding to it the faces
of Bk+t+1(p) that share an edge with ∂Bk+t(p). We claim that B is a disk. If B is not a
disk, then B is obtained as follows. Let B′ be the disk obtained from Bk+t(p) by adding to
it a diamond necklace of faces. Now B is obtained from B′ by identifying edges or vertices
on ∂B′. Note that it is not possible to identify faces because ∂Bk+t(p) is a cycle.
It is not possible to identify two non-consecutive edges of ∂B′ because any two such
edges on ∂B′ contain endpoints that are on ∂Bk+t(p), which is a disk, and therefore are
distinct. Now suppose that e1 and e2 are consecutive edges on ∂B′ whose common end-
point is v. The degree of v in B′ is either 2 or 4. It is not possible that v has degree 2 in B′
because identifying e1 and e2 would then yield a vertex of degree 1 in G. It is not possible
that v has degree 4 in B′ because identifying e1 and e2 would create a vertex of degree 3
in Bk+t(p) that is not among p, v1, . . . , vn.
So it must be that B is obtained from B′ by identifications of vertices along ∂B′. Note
that the degree in B′ of the vertices of ∂B′ alternate between 2 and 4 where the degree-2
vertices have distance k+t+2 from p and the degree-4 vertices have distance k+t+1 from
p. Thus identification of any two degree-2 vertices on ∂B′ will then force the identification
of another pair of degree-2 vertices on ∂B′. These identifications will continue until we
force G to contain either a facial cycle of length 2 (a contradiction) or a vertex of degree 2
(again a contradiction).
Thus B′ = B is a disk and the vertices of ∂B alternate with distances k + t + 1 and
k+ t+2 from p. Thus B = Bk+t+1(p) because if there is a face f in Bk+t+1(p) but not in
B, its closest vertex to p has distance at least k+ t+1 and so this face is not in Bk+t+1(p),
a contradiction.
Proposition 4.4. If Bk+t(p) is a disk in a spherical quadrangulation G as given in Proposi-
tion 4.3, Bk+t(p) contains no curvature vertices of G other than p, v1, . . . , vn, but
Bk+t+1(p) contains an additional curvature vertex of G, then Bk+t+1(p) is also a disk
as given in Proposition 4.3, contains curvature vertices u1, . . . , un on its boundary cycle,
and each ui has distance k + t+ 2 from p.
Proof. This follows from Proposition 4.3 and the fact that the only vertices in Bk+t+1(p)
that are not in Bk+t(p) are the outer vertices of the new diamond-necklace layer of the
thorax.
Proposition 4.5. If Bk+t(p) is a disk as given in Proposition 4.4 which contains curvature
vertices p, v1, . . . , vn in its interior and curvature vertices u1, . . . un on its boundary, then
G is obtained from Bk+t(p) by identifying ∂Bk+t(p) with the boundary of a standard k-disk
D such that u1, . . . , un are identified with the endpoints of the central rays of D.
Proof. By Proposition 4.4, each ui has distance k+t+1 from p. This implies that k+t+1
is even, and since k is even, we must have that t is odd.
Say that l is the smallest distance in G from the pole q to any vertex on ∂Bk+t(p) and
let u be such a vertex. It must be that d(u, p) = k + t + 1 rather than k + t, because the
vertices of ∂Bk+t(p) of distance k + t from p are saturated by Bk+t(p) and so any path
from q to one of these vertices of distance k + t from p must go through the vertices of
distance k + t + 1 from p. Therefore u is black and has degree 2 in Bk+t(p). Also, since
u is black, l must be even.
First suppose that u can be chosen to be in Bl−1(q). The intersection of Bl−1(q) and
Bk+t(p) may only consist of a collection of black vertices because the white vertices of
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 347
Bl−1(q) have distance at most l − 1 from q. Therefore Bl−2(q) contains no curvature
vertices aside from q and Proposition 4.1 implies that Bl−1(q) is a standard (l − 1)-disk.
Thus u has degree 2 in Bl−1(q) and must have degree 4 in G. The neighbor wp of u on
∂Bk+t(p) in the direction of rotation is saturated by Bk+t(p) and the neighbor wq of u
on ∂Bl−1(q) in the rotational direction is either saturated or, alternatively, is the endpoint
of a central ray and, because it is white, has codegree 1 with respect to Bl−1(q). Let
up be the next vertex in rotational order along ∂Bk+t(p) and let uq be the next vertex in
rotational order along ∂Bl−1(q). If wq is saturated by Bl−1(q), then because every face of
G has length four we must have that up = uq and this vertex has degree 4 in G. (See the
left configuration in Figure 18). If wq is the end of a central ray of Bl−1(q) (which has
codegree 1), then again, the fact all faces have length four implies that up is adjacent to wq
and so up is a curvature vertex of G and uq = u′p where u
′
p is the next black vertex in the
rotational direction on ∂Bk+t(p). (See the right configuration in Figure 18).
u
wp
wq
up
uq
Face of
length 4 u
wp
wq
up Face of
length 4
uq
Central
Ray
w′p
u′p
Figure 18: Forced vertex identifications on the boundaries of Bk+1(p) and a standard k-
disk.
This process of identifying vertices and adjacencies continues all the way around
∂Bk+t(p) and ∂Bl−1(q) so that the black vertices on ∂Bk+t(p) correspond to the black
vertices and endpoints of the central rays of Bl−1(q). Therefore l = k and G is obtained as
stated in the proposition.
Next suppose that u cannot be chosen to be in Bl−1(q). Proposition 2.6 implies that
u is the endpoint of a central ray of Bl(q). Let w be this endpoint of the central ray of
Bl−1(q) that is adjacent to u. Thus u has codegree 1 or 2 with respect to Bk+t(p). In either
case there is a face f incident to the uw-edge that contains an edge wb1 of ∂Bl−1(q) and
an edge uw1 of ∂Bk+t(p). So now a fourth edge for f would be w1b1; however, w1 is
saturated by Bk+t(p) and b1 /∈ Bk+t(p), a contradiction (see Figure 19).
Theorem 4.6. The graph described in Proposition 4.5 is given by the Three-Parameter
Construction.
Proof. Consider the part of Bk(p) between two consecutive central rays, call it Wk. Let
o1 and o2 be the curvature vertices on the central rays of Wk which have distance k from
p. Consider the black diagonal line D in Wk from o1 to o2. Now let W be the portion
of Bk+t(p) consisting of Wk along with the faces between the black diagonals emanating
from o1 and o2 which are perpendicular to D. Let o be the curvature vertex in W which
348 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
u
w b1 /∈ Bk+t(p)
w1
f
Figure 19: Final contradiction in the proof of Proposition 4.5.
has distance k + t from p. As shown in Figure 20, there is a special integer quadrilateral
for the Three-Parameter Construction contained within W .
p
o1
o
o2
Figure 20: A special integer quadrilateral within a single wedge.
By inspection, the spherical quadrangulation constructed by the special integer quadrilat-
eral from Figure 20 contains Bk+t(p) with curvature vertices positioned as shown. By
Proposition 4.5, there is only one spherical quadrangulation which contains Bk+t(p) with
curvature vertices in a given position. Thus G is given by the Three-Parameter Construc-
tion.
4.4 Curvature vertices off the central rays
Let Bk(p) be a standard k-disk in G as described in Proposition 4.2(2). The disk Bk(p)
contains at least one orbit of n degree-3 curvature vertices. Let t ≥ 0 be the smallest integer
for which Bk+t(p) contains both orbits of n degree-3 curvature vertices. In Proposition 4.8
we describe three possible structures for Bk+t(p). Finally, we show that each structure is
given by the Four-Parameter Construction.
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 349
4.4.1 Two new types of disk
Note that the length of ∂Bk(p) is 2nk. The black vertices along ∂Bk(p) have degree 2 in
Bk(p) and the white vertices along ∂Bk(p) have degree 4 in Bk(p) save for the endpoints
of the central rays which have degree 3 in Bk(p). Consider vertices w1, v1, . . . , wn, vn in
clockwise order around ∂Bk(p) in which w1, . . . , wn are the ends of the central rays and
v1, . . . , vn is one orbit of black vertices on ∂Bk(p). Say that the distance from wi to vi
along ∂Bk(p) is 2l − 1. Let T be a t-layered stack of (k, l)-zig-zag necklaces. Note that
∂T consists of two cycles. Let ∂inT be the inner cycle of ∂T and say that the necklace
along ∂inT has a black diagonal and label these black vertices as b0, . . . , bn(k+1)−1. Note
that bj for j not divisible by k + 1 appears on ∂inT and has degree four in T except when
j = i(k + 1) + l, in which case bj has degree 3 in T . Also, the white vertices on ∂inT all
have degree 2 in T save for the white vertices on ∂inT adjacent to bi(k+1)’s, which have
degree 3 in T . Thus we can identify ∂Bk(p) with ∂inT so that vi is identified with bi(k+1)+l
and wi is adjacent to bi(k+1). We call the resulting disk Zk,l,t(p). Our discussion assumes
that t ≥ 1, but as a convention we can define Zk,l,0(p) to be the standard k-disk with l
defined by either one of the two orbits of degree-3 curvature vertices on ∂Bk(p). Note that
Zk,l,t(p) is a (k + t)-ball centered at p and every vertex v ̸= p in the interior of Zk,l,t(p)
has degree 4 in Zk,l,t(p) save for v1, . . . , vn which all have degree 3. Figure 21 depicts
Z5,2,3(p) for n = 5 (ignore the shading in the outer faces for the moment).
Figure 21: The disk Z5,2,3(p). If the shaded faces are removed, then the remaining faces
define Z5,2,2(p).
Now for t ≥ 2 that is even we define a disk Ẑk,l,t(p) from Zk,l,t(p). Say that v′i is the
black vertex on ∂Zk,l,t(p) that is on the transverse path from vi emanating outwards from
Bk(p). (Call this transverse path from vi a curvature ray.) Also, say that w′i is the endpoint
of the central ray of Zk,l,t(p) that contains wi. Now since t is even, the black vertices on
∂Zk,l,t(p) all have degree 2 except for v′1, . . . , v
′
n, which have degree 3 in Zk,l,t(p). Let
l′ = min{l − 1, k − l}. Label the l′ black vertices on ∂Zk,l,t(p) in the clockwise direction
from v′i with 1, 2, . . . , l
′ and do the same for the l′ black vertices on ∂Zk,l,t(p)
350 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
in the counter-clockwise direction from v′i (see the left-hand configuration in Figure 22 in
which k = 7, l = 3, t = 2, and l′ = l − 1 = 2).
1 12 2
vi
v′i
p
vi
p
v′i
Figure 22: Constructing Ẑk,l,t(p) with k = 7, l = 3, t = 2, and l′ = l − 1 = 2.
Now identify the black vertices having the same labels as shown on the right in Fig-
ure 22. Repeat these identifications for each i. The resulting disk is Ẑk,l,t(p). Note that
Ẑk,l,t(p) is a (k+t)-ball centered at p and that every vertex v ̸= p in the interior of Ẑk,l,t(p)
has degree 4 in Ẑk,l,t(p) save for v1, . . . , vn, v′1, . . . , v
′
n which all have degree 3.
4.4.2 The three structures
Proposition 4.7. Let Bk(p) be a standard k-disk in G with k odd and with exactly n degree-
3 vertices v1, . . . , vn of G appearing on ∂Bk(p). For each t ≥ 0, if the only curvature
vertices in Bk+t−1(p) are among p, v1, . . . , vn, then Bk+t(p) is either Zk,l,t(p) or Ẑk,l,t(p)
where l is specified by the position of v1, . . . , vn on ∂Bk(p).
Proof. The proof will be by induction on t where the case for t = 0 is given by Propo-
sition 4.1. Assuming for some t ≥ 1 that the only curvature vertices in Bk+t−1(p)
are among p, v1, . . . , vn we now consider Bk+t(p). For t = 1, we already know that
Bk+t−1(p) = Bk(p) is a standard k-disk which is also Zk,l,0(p). For t ≥ 2, the induction
hypothesis assumes that Bk+t−1(p) is either Zk,l,t−1(p) or Ẑk,l,t−1(p). However, while the
only curvature vertices of Bk+t−1(p) are among p, v1, . . . , vn, in fact, Ẑk,l,t−1(p) contains
more curvature vertices than this. Hence Bk+t−1(p) = Zk,l,t−1(p).
By Proposition 2.4 every face of G not in Bk+t−1(p) but sharing an edge with
∂Bk+t−1(p) is in Bk+t(p). Consider the face-connected subsurface B ⊆ Bk+t(p) con-
sisting of Bk+t−1(p) along with the faces not in Bk+t−1(p) but sharing an edge with
∂Bk+t−1(p). We will show that B = Zk,l,t(p) or B = Ẑk,l,t(p) and that B = Bk+t(p).
Given an edge e of ∂Bk+t−1(p) = ∂Zk,l,t−1(p), let fe be the face of B that is not in
Bk+t−1(p) and is incident to e. For comparison as a standard model, consider the disk
Zk,l,t(p) (separate from G) whose subdisk Zk,l,t−1(p) is identified with Bk+t−1(p) in G.
Let f ′e be the face of Zk,l,t(p) that is not in Bk+t−1(p) = Zk,l,t−1(p) and is incident to e.
If fe (or f ′e) is incident to a central ray, then call fe (or f
′
e) a radial face; otherwise, call fe
(or f ′e) a notch face. Note that f
′
e1 = f
′
e2 if and only if f
′
e1 is a notch face with e1 and e2
both incident to a common vertex that is saturated with respect to Bk+t−1(p) and G (see
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 351
Figure 21). We must show that the corresponding necessary and sufficient condition holds
for fe1 = fe2 .
If fe is a notch face, then fe shares two consecutive edges (say e1 and e2) with
∂Bk+t−1(p) where the common endpoint of e1 and e2, call it v12, is a vertex saturated
by Bk+t−1(p) in G. It cannot be that a third edge of fe is on ∂Bk+t−1(p) because such
an edge would have to be consecutive with e1 or e2 on the cycle ∂Bk+t−1(p), whereas the
two endpoints of e1 and e2 other than v12 have codegree 1 or 2 with respect to Bk+t−1(p).
If fe is a radial face of B, then denote the edges of fe by e1, e2, e3, e4 in rotational
order along fe. Assuming that e1 is on ∂Bk+t−1(p) and is incident to a central ray of
Bk+t−1(p) (call it r) we get that each endpoint of e1 has positive codegree with respect to
Bk+t−1(p). Thus e2 and e4 are not on the cycle ∂Bk+t−1(p). Without loss of generality
assume that edge e2 is the transverse continuation of r. Assume by way of contradiction
that e3 is on ∂Bk+t−1(p). Since e2 and e4 are not on ∂Bk+t−1(p), we again must have that
e3 is also incident to a central ray of Bk+t−1(p), call it r′. Note that r ̸= r′ because r = r′
would imply that G is not simple, a contradiction. Now either e1 and e3 are in the same
orbit under the rotational symmetry or not. If so, then the orbit of fe under the rotational
symmetry consists of n/2 faces and so there is a pseudofixed point in the interior of fe, a
contradiction. If not, then when adding fe to Bk+t−1(p), the black and white bipartition
forces there to be a half twist which creates a Möbius band in G, a contradiction.
The previous two paragraphs show that fe 7→ f ′e is a one to one correspondence be-
tween the faces of B that are not in Bk+t−1(p) and the faces of Zk,l,t(p) that are not in
Bk+t−1(p). Furthermore, fe 7→ f ′e takes notch faces to notch faces and radial faces to
radial faces; also, if f1 and f2 are two consecutive faces of B, then their common vertex
along ∂Bk+t−1(p), call it v, has codegree one or two and this determines whether or not
f1 and f2 share an edge incident to v. Therefore B is obtained from Zk,l,t(p) by making
zero or more identifications along ∂Zk,l,t(p). If there are no identifications, then we have
that B = Zk,l,t(p). We also get that B = Bk+t(p) because no face outside of B can have
a vertex of distance k + t− 1 from p and so we are done. So now, in Case 1 say that there
are edges on ∂Zk,l,t(p) that are identified and in Case 2 say that no edges along ∂Zk,l,t(p)
are identified but that there are vertices that are identified.
Case 1: Assume that e1 and e2 are on ∂Zk,l,t(p) and are identified in going from Zk,l,t(p)
to B. If ei is not incident to a central ray of Zk,l,t(p), then ei has one endpoint that is on
Bk+t−1(p) = Zk,l,t−1(p), is not a curvature vertex, and has degree 4 in Zk,l,t(p). Any
identification with another vertex of the same color would yield a vertex of degree more
than four, a contradiction. Thus e1 and e2 are both incident to central rays of Zk,l,t(p).
Because the rotational symmetry has only two fixed points (i.e., the poles), the n endpoints
of the central rays of Zk,l,t(p) must either correspond to n distinct vertices in B or one
vertex in B that is fixed under the n-fold rotational symmetry (that is, the other pole of
the rotational symmetry, call it q). We assume the latter is true as this is the only way in
which edges of ∂Zk,l,t(p) may be identified. Now the 2n edges of ∂Zk,l,t(p) incident to
the central rays are identified as per the numbering in Figure 23 to obtain Z ′.
There are two subcases to consider here: in Case 1.1 say that 2 ≤ l ≤ k − 1 and in
Case 1.2 say that l ∈ {1, k}.
Case 1.1: Now ∂Z ′ consists of n vertex-disjoint cycles. Since q is black, the black vertices
on ∂Z ′ all have degree 4 in Z ′ and the white vertices on ∂Z ′ all have degree 2 in Z ′ except
352 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
q q
q
q
q
p
1
2
3
2
3
4
45
1
5
q5
q
q
q
q
4
4
3
3
2
2
1
1
5
Figure 23: The disk Z ′ for l = 2 and l = 1, respectively.
for exactly 2n white vertices on ∂Z ′ having degree 3 in Z ′: the n endpoints of the curvature
rays along with the n neighbors of q. As shown at the beginning of Case 1, no two edges
of ∂Z ′ may be identified and so B is obtained from Z ′ by the identification of zero or more
pairs of white vertices having degree 2 in Z ′ to obtain white vertices of degree 4 in G.
We cannot, of course, identify white vertices from two distinct cycles of ∂Z ′ because this
would create a non-separating cycle in the embedding of G in the sphere, a contradiction.
Consider a cycle C in ∂Z ′ with vertices w1, b1, w2, b2, . . . , wm, bm and say by way of
contradiction that wi and wj are identified in going from Z ′ to B. Call the resulting face-
connected subsurface after this identification Z ′′. Now wi, bi, bj−1 (and also wi, bi−1, bj)
all have degree four in Z ′′. Because 2 ≤ l ≤ k − 1, we now get that these three vertices
are on a common face of G and so wi+1 and wj−1 (and also wi−1 and wj+1) must be
identified in going from Z ′′ to B. This identification process will continue and eventually
yield a contradiction by either: trying to identify a white vertex of degree 3 with a white
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 353
vertex of degree 2 or 3, by creating a face of length four having a white vertex of positive
codegree on its boundary, or by creating a cycle of length 2.
Thus we must have that Z ′ = B; however, this is not possible for the following reason.
Consider a cycle C in ∂B and let P be the disk of G with ∂P = C. The black vertices
on C have degree 2 in P , the white vertices on C have degree 4 in P save for two of them
which have degree 3 in P . By the rotational symmetry, the interior vertices of P include
exactly one black degree-3 vertex and all other interior vertices have degree 4 in P . Thus
P has an odd number of vertices of odd degree, a contradiction.
Case 1.2: As in Case 1.1, ∂Z ′ consists of n vertex-disjoint cycles and the black vertices
on ∂Z ′ all have degree 4 in Z ′. However, in this case, the white vertices on ∂Z ′ all have
degree 2 in Z ′ except for the n white vertices adjacent to q which have degree 4 in Z ′. Let
C be one cycle in ∂Z ′. As in Case 1.1, no two edge of C may be identified in going from
Z ′ to B. Label the vertices in rotational order along C with w1, b1, . . . , wm, bm where w1
is the white vertex of C having degree 4 in Z ′. Since bm, w1, b1 all have degree 4, they
must be on the same face of G and so we must have that wm = w2. This in turn implies
that wm−1 = w3, etc. These identifications are not possible, however, because m = 2k−2
and k is even and thus these identifications would create a face of length 2, a contradiction.
Case 2: In this case, the only possible identifications along ∂Zk,l,t(p) in going from
Zk,l,t(p) to B are pairs of vertices which have degree 2 in Zk,l,t(p). Say that the ver-
tices on ∂Zk,l,t(p) of distance k + t + 1 from p have color κ and the vertices of distance
k + t have color λ. Hence {κ, λ} = {black,white} (e.g., in Figure 21 κ = white). The
vertices of color κ on ∂Zk,l,t(p) have degree 2 in Zk,l,t(p) save for the n endpoints of the
curvature rays (which have degree 3 in Zk,l,t(p)) and the vertices of color λ on ∂Zk,l,t(p)
have degree 4 in Zk,l,t(p) save for the n endpoints of the central rays (which have degree 3
in Zk,l,t(p)).
Label the vertices along ∂Zk,l,t(p) in rotational order with λ1, κ1, . . . , λm, κm in which
λ1 is the endpoint of a central ray. Say that κi and κj are identified in B, and say that
v1, . . . , vn is the orbit of κi under the rotational symmetry and u1, . . . , un the orbit of κj .
Then |{u1, . . . , un, v1, . . . , vn}| = 2n in Zk,l,t(p) and |{u1, . . . , un, v1, . . . , vn}| = n or 1
in G. It cannot be that |{u1, . . . , un, v1, . . . , vn}| = 1 in G because then these 2n degree-2
vertices in Zk,l,t(p) would then identify to one vertex of degree 4n in G, a contradiction.
Thus |{u1, . . . , un, v1, . . . , vn}| = n in G. Because of the rotational symmetry these two
orbits of vertices must alternate along the cycle ∂Zk,l,t(p) and because G is spherical,
identified pairs of vertices (e.g., κi and κj) must appear consecutively along ∂Zk,l,t(p).
Let γij be the κiκj-path along ∂Zk,l,t(p) which contains no other vertices from
v1, . . . , vn, u1, . . . , un. Again, because of the rotational symmetry, at most one endpoint
of a curvature ray and at most one endpoint of a central ray occurs on γij . Suppose that
λi+1 and λj are the neighbors of κi = κj on γij . If λi+1 and λj both have degree 4 in
Zk,l,t(p), then κi, λi+1, and λj must all be on the same face of G and so we must have
that κi+1 = κj−1 in B. Similarly if λi and λj+1 both have degree 4 in Zk,l,t(p), then
κi−1 = κj+1 in B. These identifications of degree-2, κ-colored vertices must continue
in each direction along ∂Zk,l,t(p) until either we reach the endpoint of a curvature ray or
central ray. Thus γij contains either the endpoint of a curvature ray or the endpoint of a
central ray, but not both.
We claim that γij contains the endpoint of a curvature ray and not the endpoint of a
central ray. This is because if the latter were true, then, because the endpoint of a central
354 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
ray is of color λ, this identification of κ-colored vertices along γij would end with either a
face of length two (a contradiction) or a face of length four containing a κ-colored vertex
of degree 2 (again a contradiction).
Now since the endpoint of a curvature ray is contained in γij , the identification of κ-
colored vertices along γij ends with the identification of some κa−1 and κa+1 where κa
is the endpoint of the curvature ray and so has degree 3 in G. Also note that this implies
that κa is the midpoint of γij . Identifications of κ-colored vertices from κi = κj that
are off of γij must then stop at the endpoint of the central rays. These identifications of
vertices in Zk,l,t(p) result in the disk Ẑk,l,t(p). These are the only vertex identifications
that can happen in going from Zk,l,t(p) to B because we started with an arbitrary vertex
identification. Thus B = Ẑk,l,t(p) and we must also have that B = Ẑk,l,t(p) = Bk+t(p)
because any face of G outside of Ẑk,l,t(p) cannot contain a vertex of distance k + t − 1
from p.
Proposition 4.8 gives us the three possible structures for Bk+t(p). The proof of Propo-
sition 4.8 is similar to the proof of Proposition 4.2 using Proposition 4.7 in the place of
Proposition 4.1.
Proposition 4.8. Let k ≥ 1 be odd and let O1 be one orbit of n degree-3 curvature vertices
contained in ∂Bk(p). Let O2 be the second orbit of n degree-3 curvature vertices of G. Let
t ≥ 0 be such that Bk+t−1(p) contains no curvature vertices aside from O1 ∪ {p} and
Bk+t(p) contains O2. One of the following holds.
(1) If t is odd (that is, k+ t is even), then Bk+t(p) = Zk,l,t(p) and the vertices of O2 are
the endpoints of the central rays of Bk+t.
(2) If t is even (that is, k + t is odd), then either
(a) Bk+t(p) = Ẑk,l,t(p) and the vertices of O2 are the endpoints of the curvature
rays and appear in the interior of Bk+t(p), or
(b) Bk+t(p) = Zk,l,t(p) and the vertices of O2 are on ∂Bk+t(p) but not the end-
points of the central rays or curvature rays.
Proposition 4.9. If Bk+t(p) = Zk,l,t(p) is as given in Proposition 4.8(2)(b), then G is
given by the Four-Parameter Construction with uniquely determined parameters.
Proof. Consider two consecutive central rays of Zk,l,t(p) along with the vertices, edges and
faces of Zk,l,t(p) between these central rays, that is, the closure of a fundamental region of
the rotational symmetry, call it F . A rendering of F is shown on the left in Figure 24 in
which the dashed lines have length t (with t = 0 a possibility) and are identified.
Let o1 ∈ O1 and o2 ∈ O2 be the vertices of O1 ∪O2 in F . Since o2 is not on the endpoint
of the central ray or curvature ray, o2 appears on ∂Zk,l,t(p) in one of the two circled areas
shown on the left of the figure; take the right of Figure 24 as an illustration. We may assume
without loss of generality that o2 is in the upper region because if o2 is in the lower circled
region on the left of Figure 24, then we may reflect Zk,l,t(p) to get Zk,k−l+1,t(p) and then
have o2 in the upper region.
Now take the fundamental region F ′ adjacent to and in the counterclockwise direction
from F . The rendering of F ∪ F ′ shown in Figure 25 is geometrically flat and so we coor-
dinatize in the obvious way with p at (0, 0). As a result, the grey lines shown in Figure 25
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 355
p
o1
p
o1
o2
Figure 24: The fundamental region F .
form a quadrilateral with a right angle at the origin. Evidently the quadrilateral is also con-
vex because each of the interior angles is less than 180◦. Hence the grey quadrilateral is
of the type used in the Four-Parameter Construction; furthermore, the parameters defining
this quadrilateral are uniquely determined by the positions of o1 and o2 within F . Hence
Zk,l,t(p) contains n of these special quadrilaterals at p.
o′1
o′2
p
o1
o2
F ′
F
Figure 25: Two fundamental regions F and F ′ along with a four-parameter special integer
quadrilateral.
Conversely, by Proposition 4.7, the entire Four-Parameter Construction using this
quadrilateral must contain Zk,l,t(p) because the positions of the curvature vertices in the
Four-Parameter Construction come from the corners of the quadrilaterals. So one possi-
bility for G is given by the Four-parameter construction. Now, we will show that there is
only one spherical quadrangulation which contains Zk,l,t(p) and has O2 in this position on
∂Zk,l,t(p), which will complete our proof.
Consider the vertices on the boundary ∂Zk,l,t(p). The black vertices on this boundary
356 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
have degree 2 in Zk,l,t(p) save for the n endpoints of the curvature rays which have degree
3. The white vertices on this boundary have degree 4 in Zk,l,t(p) save for the n endpoints of
the central rays which have degree 3. Let w1, b1, c1, . . . , wn, bn, cn be vertices in clockwise
rotational order on ∂Zk,l,t(p) in which the wi’s are the endpoints of the central rays, bi’s
the vertices of O2, and ci’s the endpoints of the curvature rays.
Now let D be the disk of G constructed from the faces of G not in Zk,l,t(p). Thus D
contains the pole q ̸= p and ∂D = ∂Zk,l,t(p); furthermore, by rotational symmetry the
pole q is in the interior of D. Now,
• the wi’s, bi’s, and ci’s all have degree 3 in D,
• the remaining black vertices on ∂D have degree 4 in D, and
• the remaining white vertices on ∂D have degree 2 in D.
By Proposition 2.8, we may consider D as a subgraph of a standard r-disk Sr with a
black central vertex, call it q0, for some large enough value of r. Of course the embedding
of D must have q corresponding to q0 and, since q is in the interior of D, the central rays
of D must lie on the central rays of Sr. Thus the embedding of D in Sr is unique up to
dihedral symmetry. For the uniqueness of D as a completion of G, we need to show that
there is no other disk D′ in Sr having n-fold rotational symmetry around q with a bijection
between the vertices of ∂D′ and ∂D which respects degrees and cyclic ordering.
Consider the black vertices of Sr and connect pairs of black vertices on the same face
with an edge (say it is also black). This black graph is a quadrangulation with every internal
vertex of degree 4 aside from q which has degree n. Call any transverse path in the black
graph a black diagonal path of Sr or D. Call the n black diagonal paths of Sr or D that
originate from q the diagonal rays of Sr or D.
Now consider the boundary faces of D and the black diagonal edges in each. These
black edges form a cycle, call it C, in the black-diagonal graph and C is contained entirely
inside the disk D. Note that the cycle C consists of black diagonal paths whose endpoints
are the ci’s, bi’s, and w′i’s where w
′
i is the black neighbor of wi that is not in Zk,l,t(p).
Traversing C in Sr with q to our right, the ci’s and bi’s represent a right turn rather than a
transverse path and the w′i’s represent a left turn. We will now show that C (and hence the
boundary faces of D) is uniquely determined by the positions of ci’s, bi’s, and w′i’s on ∂D.
This will imply the uniqueness of D.
Now let V be the region of Sr between and including two consecutive diagonal rays,
call them Y1 and Y2 in the clockwise direction. The intersection of C with Y1 has one or
more connected components, each of which is either an isolated vertex or a path of positive
length. If there is no path of positive length, then let y1 be some vertex of C on Y1. If
there is a path of positive length in the intersection, then let y1 be the last vertex of some
intersection path when traversing C in the clockwise direction. Let y2 be the corresponding
vertex on Y2 under the rotational symmetry in the clockwise direction, and let P be the
y1y2-path in C in the clockwise direction. Consider the square Q in V given by the black
diagonals of Sr shown in Figure 26.
In the clockwise traversal of C, P contains two right turns and one left turn and at the rest
of the vertices of P , a transverse crossing. The sequence of turns is either left-right-right,
right-left-right, or right-right-left; however, if necessary we can reflect R around the axis
Y1 and reverse the traversal of C so that the first turn is right. In Figure 26, V is rendered
as part of the standard 4 × 4 grid in the xy-plane between the perpendicular lines y = x
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 357
Y2Y1
y2
y1
q
q0
Figure 26: The square defined by Y1, y1, Y2, and y2 in V .
and y = −x. In the traversal of P , two right turns and a left turn yield a net change of 90◦
degrees in the clockwise direction. Because of the way in which y1 is chosen, the first edge
of P is in the direction of the arrow shown in Figure 26. Now the path P is in V and is
completely determined by the placement of the two right turns in the 3-turn sequence. In
Figure 27 we have three examples of P . Since these turns are determined by the placements
of the ci’s, bi’s, and wi’s on ∂D, there is only one possibility for P and so for C and hence
for D.
Proposition 4.10. If Bk+t(p) = Ẑk,l,t(p) is as given in Proposition 4.8(2)(a), then G is
given by the Four-Parameter Construction with uniquely determined parameters.
Proof. As in the proof of Proposition 4.9, consider two adjacent fundamental regions F
and F ′ of Ẑk,l,t(p) between three consecutive central rays. These may be rendered in a
geometrically flat fashion as in Figure 28 with identically labeled vertices being identified
in G and appropriate identifications of dashed edges. Note that the path of dashed edges
has positive length. Since the rendering is flat we have a special integer quadrilateral as
used in the Four-Parameter Construction with parameters uniquely determined by k, l, and
t as shown in the figure. Therefore the Four-Parameter Construction yields one possibility
for G. In order to show that this is the only possibility for G, we will show that there is
only one possibility for the disk in G around q sharing its boundary with Ẑk,l,t(p).
Consider the dashed edge shown in Figure 29 along with its orbit of n edges under the
rotational symmetry. Let Z̃ be the disk around p consisting of Ẑk,l,t(p) along with the n
faces bounded by these n edges and Ẑk,l,t(p).
Let D be the disk defined by the faces of G not contained in Z̃. Note that D contains q
in its interior and ∂D = ∂Z̃. All of the white vertices of ∂D = ∂Z̃ have degree 4 in Z̃ and
degree 2 in D. Among the black vertices of ∂D = ∂Z̃, n have degree 3 in both Z̃ and D
and the rest have degree 2 in Z̃ and degree 4 in D.
Say that l is the smallest distance in G from the pole q to any vertex on ∂D = ∂Z̃. Let
u be one such vertex on the common boundary. It must be that d(u, p) = k + t+ 1 rather
358 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
q
q0
q
q0
q
q0
Figure 27: Three examples of the path P . In the third example, B intersects Y1 in a path.
than k + t. This is because the vertices of ∂Bk+t(p) of distance k + t from p are all white
and are saturated by Z̃ and so any path from q to one of these vertices of distance k + t
from p must go through the vertices of distance k+ t+ 1 from p. Therefore u is black and
has degree 2 or 3 in Z̃.
Since u is black, l must be even. First suppose that u can be chosen to be in Bl−1(q)
(which by Proposition 4.1 is a standard (l−1)-disk); that is, u is a vertex of degree 2 on the
boundary of Bl−1(q). Since the white vertices of ∂Bl−1(q) have distance l − 1 from q and
l is the smallest distance of a vertex from q to Z̃, these white vertices on ∂Bl−1(q) are not
in Z̃. So any black vertex on ∂Bl−1(q) which is identified to a black vertex on ∂Z̃ forces
another identification of two black boundary vertices. Eventually these identifications will
run to the degree-3 vertices of Z̃ on ∂Z̃. But this forces these black vertices to have degree
5 in G, a contradiction.
L. Abrams and D. Slilaty: Characterization of a family of rotationally symmetric spherical . . . 359
o2
o′2
o′2
p
o1
o′1
b(1,1)
o′1
b(1,1)
b(1,2)
b(1,2)
b(2,1)
b(2,1)
b(2,2)
b(2,2)
Figure 28: A flat rendering of two fundamental regions for Ẑk,l,t(p).
vi
p
v′i
Figure 29: There are n edges in G with endpoints in Ẑk,l,t(p). One such edge is shown as
a dashed curve.
Thus u is a vertex of degree 3 on the boundary of Bl(q); that is, u is the endpoint of a
central ray of Bl(q) which by Proposition 4.1 is a standard l-disk. Degree considerations
now force u and the vertices in its orbit to be identified with the degree-3 vertices of Z̃ on
∂Z̃. From here ∂Bl(q) must then be identified with ∂Z̃.
Proposition 4.11. If Bk+t(p) = Zk,l,t(p) is as given in Proposition 4.8(1), then G is given
by the Four-Parameter Construction with uniquely determined parameters.
Proof. Again, as in the proof of Proposition 4.9, consider two adjacent fundamental regions
F and F ′ of Zk,l,t(p) between three consecutive central rays rendered in a geometrically
flat fashion as in Figure 30. Again we have a special integer quadrilateral contained in
F ∪ F ′ as shown in the figure with uniquely determined parameters. Therefore the Four-
Parameter Construction yields one possibility for G, and we will now show that there is
only one possibility for the disk in G around q sharing its boundary with Zk,l,t(p).
Let m be the shortest distance from q to a vertex u on ∂Zk,l,t(p). Since all of the black
vertices on ∂Zk,l,t(p) are saturated by Zk,l,t(p), it must be that u is white and hence m is
odd. By Proposition 4.1 and the definition of m, Bm(q) is a standard m-disk. If u is not
the endpoint of a central ray of Bm(q), then u is saturated by Bm(q). Since u has degree
360 Ars Math. Contemp. 22 (2022) #P2.10 / 327–361
o2
p
o1
o′1
o′2 o
′
n
Figure 30: A flat rendering of two fundamental regions for Zk,l,t(p).
4 in G, it must be that u has degree 2 in ∂Zk,l,t(p) and that the boundary edges incident
to u in Bm(q) are identified to the boundary edges incident to u in ∂Zk,l,t(p). The only
boundary edges of both disks that are left are those incident to the central rays of Bm(q)
and the curvature rays of ∂Zk,l,t(p). Degree considerations and the fact that all faces must
have length 4 now force all edges of ∂Bm(q) to be identified to all edges of ∂Zk,l,t(p).
ORCID iDs
Lowell Abrams https://orcid.org/0000-0002-8174-5957
Daniel Slilaty https://orcid.org/0000-0002-7918-3641
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include the full postal address and the country name. If avilable, specify the e-mail
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