ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 14 (2018) 1-14
https://doi.org/10.26493/1855-3974.1088.414
(Also available at http://amc-journal.eu)
Octahedral, dicyclic and special linear solutions of some Hamilton-Waterloo problems
Simona Bonvicini
Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universita di Modena e Reggio Emilia, Via Campi 213/A, 41125 Modena, Italy
Marco Buratti
Dipartimento di Matematica e Informatica, Universita degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
Received 21 April 2016, accepted 23 August 2016, published online 22 March 2017
We give a sharply-vertex-transitive solution of each of the nine Hamilton-Waterloo problems left open by Danziger, Quattrocchi and Stevens.
Keywords: Hamilton-Waterloo problem, group action, octahedral binary group, dicyclic group, special linear group.
Math. Subj. Class.: 05C70, 05E18, 05B10
1 Introduction
A cycle decomposition of a simple graph r = (V, E) is a set D of cycles whose edges partition E. A partition F of D into classes (2-factors) each of which covers all V exactly once is said to be a 2--factorization of r. The type of a 2-factor F is the partition n = [IT,..., i" ] (written in exponential notation) of the integer |V | into the lengths of the cycles of F.
A 2-factorization F of Kv (the complete graph of order v) or Kv -1 (the cocktail party graph of order v) whose 2-factors are all of the same type n is a solution of the so-called Oberwolfach Problem OP(v; n). If instead the 2-factors of F are of two different types n and then F is a solution of the so-called Hamilton-Waterloo Problem HWP(v; n, r, s) where r and s denote the number of 2-factors of F of type n and respectively.
A complete solution of the OPs whose 2-factors are uniform, namely of the form OP(^n; [in\), has been given in [1] and [12]. Other important classes of OPs has been
E-mail addresses: simona.bonvicini@unimore.it (Simona Bonvicini), buratti@dmi.unipg.it (Marco Buratti)
Abstract
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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Ars Math. Contemp. 14 (2018) 97-116
solved in [4, 15]. For the time being, to look for a solution to all possible OPs and, above all, HWPs is too ambitious. Anyway it is reasonable to believe that we are not so far from a complete solution of the HWPs whose 2-factors are uniform, namely of the form HWP(v; [hv/h], [wv/w]; r, s). We can say this especially because of the big progress recently done in [10].
Danziger, Quattrocchi and Stevens [11] treated the HWPs whose 2-factors are either triangle-factors or quadrangle-factors, they namely studied HWP(12n; [34n], [43n]; r,s). In the following such an HWP will be denoted, more simply, by HWP(12n; 3,4; r, s). They solved this problem for all possible triples (n, r, s) except the following ones:
(i)	(4, r, 23 - r) with r G {5, 7, 9,13,15,17};
(ii)	(2, r, 11 - r) with r G {5, 7, 9}.
Six of the nine above problems have been recently solved in [14] where it was pointed out that all nine problems were also solved in a work still in preparation [2] by the authors of the present paper. Meanwhile, a solution for each of the remaining three problems not considered in [14] have been given in [16]. Notwithstanding, in the present paper we want to present our solutions to the nine HWPs left open by Danziger, Quattrocchi and Stevens in detail. These solutions, differently from those of [14, 16], are full of symmetries since they are G-regular for a suitable group G. We recall that a cycle decomposition (or 2-factorization) of a graph r is said to be G-regular when it admits G as an automorphism group acting sharply transitively on all vertices. Here is explicitly our main result:
Theorem 1.1. There exists a O-regular 2-factorization of K48 — I having r triangle-factors and 23 — r quadrangle-factors where O is the binary octahedral group and r G {5, 7, 9,13,15,17}.
There exists a Q24-regular 2-factorization of K24 — I having r triangle-factors and 11 — r quadrangle-factors where Q24 is the dicyclic group of order 24 and r G {7,9}.
There exists a SL2(3)-regular 2-factorization of K24 — I having six triangle-factors and five quadrangle-factors where SL2(3) is the 2-dimensional special linear group over Z3.
2 Some preliminaries
The use of the classic method of differences allowed to get cyclic (namely Zv -regular) solutions of some HWPs in [8, 9, 13]. Now we summarize, in the shortest possible way, the method of partial differences. This method, explained in [7] and successfully applied in many papers (see, especially, [6]), has been also useful for the investigation of G-regular 2-factorizations of a complete graph of odd order [9]. The G-regular 2-factorizations of a cocktail party graph can be treated similarly.
Throughout this paper any group G will be assumed to be written multiplicatively and its identity element will be denoted by 1. Let Q be a symmetric subset of a group G; this means that 1 G Q and that w G Q if and only if w-1 G Q. The Cayley graph on G with connection-set Q, denoted by Cay[G : Q], is the simple graph whose vertices are the elements of G and whose edges are all 2-subsets of G of the form {g, wg} with (g, w) G G x Q.
Remark 2.1. If A is an involution of a group G, then Cay[G : G \ {1, A}] is isomorphic to K|G| — I. So, in the following, such a Cayley graph will be always identified with the cocktail party graph of order |G|.
S. Bonvicini and M. Buratti: Octahedral, dicyclic and special linear solutions of some
3
Let Cycle(G) be the set of all cycles with vertices in G and consider the natural right action of G on Cycle(G) defined by (c1,c2,..., cn)g = (c1g, c2g,..., cng) for every C = (c1,c2,... ,cn) e Cycle(G) and every g e G. The stabilizer and the orbit of any C e Cycle(G) under this action will be denoted by Stab(C) and Orb(C), respectively. The list of differences of C e Cycle(G) is the multiset AC of all possible quotients xy-1 with (x, y) an ordered pair of adjacent vertices of C. One can see that the multiplicity mAc(g) of any element g e G in AC is a multiple of the order of Stab(C). Thus it makes sense to speak of the list of partial differences of C as the multiset dC on G in which the multiplicity of any g e G is defined by
, s m-Ac (g) mac (g) :=
\Stab(C )|'
We underline the fact that dC is, in general, a multiset. Note that if dC is a set, namely without repeated elements, then it is symmetric so that it makes sense to speak of the Cayley graph Cay[G : dC]. The following elementary but crucial result holds.
Lemma 2.2. If C e Cycle(G) and dC does not have repeated elements, then Orb(C) is a G-regular cycle-decomposition of Gay[G : dC].
By Remark 2.1, as an immediate consequence of the above lemma we can state the following result.
Theorem 2.3. Let A be an involution of a group G. If {C1,... ,Ct} is a subset of Cycle(G) such that {Ji=1 dCi = G \{1,A}, then[Jt = 1 Orb(Ci) is a G-regular cycle-decomposition of K|G| -1.
We need, as last ingredient, the following easy remarks.
Remark 2.4. If C e Cycle(G) and V(C) is a subgroup of G, then Orb(C) is a 2-factor of the complete graph on G whose stabilizer is the whole G.
If C1,... ,Ct are cycles of Cycle(G) and |J t=1 V (CCi) is a complete system of representatives for the left cosets of a subgroup S of G, then |Jt=1 OrbS(Ci) is a 2-factor of the complete graph on G whose stabilizer is S.
3 Octahedral solutions of six Hamilton-Waterloo problems
Throughout this section G will denote the so-called binary octahedral group which is usually denoted by O. This group, up to isomorphism, can be viewed as a group of units of the skew-field H of quaternions introduced by Hamilton, that is an extension of the complex field C. We recall the basic facts regarding H. Its elements are all real linear combinations of 1, i, j and k. The sum and the product of two quaternions are defined in the natural way under the rules that
i2 = j2 = k2 = ijk = -1. If q = a + bi + cj + dk = 0, then the inverse of q is given by
a — bi — cj — dk
q = ■
b2 + c2 + d2 '
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Ars Math. Contemp. 14 (2018) 97-116
The 48 elements of the multiplicative group G are the following:
±1, ±i, ±j, ±k;
1 (±1 ± i ± j ± k);
7=(±x ± y), {x,y}€ ({1'V'fc}).
The use of the octahedral group G was crucial in [3] to get a Steiner triple system of any order v = 96n + 49 with an automorphism group acting sharply transitively an all but one point. Here G will be used to get a G-regular solution of each of the six Hamilton-Waterloo problems of order 48 left open in [11]. We will need to consider the following subgroups of G of order 16 and 12, respectively:
K = <k, ^(j - k)>; (
L = <7= (j - k), 1 (-1 - i + j + k)>.
3.1 An octahedral solution of HWP(48; 3,4; 5,18)
Consider the nine cycles of Cycle(G) defined as follows.
Ci = C2 = C3 = C4 = C5 = Ce = C7 = C8 =
C9 =
-7= (1 - k), 2 (1 - i - j - k))
72
1 (-1 - i + j + k), 1 (-1 + i - j - k))
1 (-1 + i + j - k), I(-1 - i - j + k)) k, -1, -k)
j, -1, -j) )
7= (-i + k), - = (1 + i + j + k), -7= (j + k)) 7= (i - j), 7= (1+i), 1 (1 - i - j + k)) 1 (1 - i+j -k), k, -7= (1+j))
7= (1 - i), -7= (1 + i), = (-1 - i+j - k))
We note that Sta6(Cj) = V(C,) for 2 < i < 5 while all other C,'s have trivial stabilizer. Thus, by Lemma 2.2, one can check that Or6(Cj) is a ij-cycle decomposition of Cay[G : «¿] where i is the length of C, and where the «¿'s are the symmetric subsets of G listed below.
«1 = {-7= (1 - k), = (1 - i - j - k), -7= (1+i)}±1
±1
±1
«= = { = (-1 - i + j + k)} «3 = { = (-1 + i + j - k)} «4={k}±1 «5 ={j}±1
«6 = {7= (-i + k), 7= (j - k) 7= (1 - k) -7= (j + k)}±1
S. Bonvicini and M. Buratti: Octahedral, dicyclic and special linear solutions of some
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{7(i - j), 1 (1 + i - j - k), 71 (i + j), 2(1 - i - j + k)}±1
Os = {2(1 - i + j - k), -1(1 + i + j + k), -7(i + k), -7(1+ j)}±1
On
{7 (1 - i),i, 7(1+ j), 2(-1 - i + j - k)}
±1
One can see that the ^'s partition G \ {1, -1}. Thus, by Theorem 2.3 we can say thatC := U9=1 Or6G(Ci) is a G-regular cycle-decomposition of K48 - I. Now set F =
OrbSi (Gj) where
{K for i = 1; G for 2 < i < 5; L for 6 < i < 9.
By Remark 2.4, each F is a 2-factor of K48 — I with Stab(F) = Si, hence Or6(F) has length 3 or 1 or 4 according to whether i = 1, or 2 < i < 5, or 6 < i < 9, respectively. The cycles of F are triangles or quadrangles according to whether or not i <3. Thus, recalling that C is a cycle-decomposition of K48 — I, we conclude that F := |J9=1 Or6(F) is a G-regular 2-factorization of K48 — I with 5 triangle-factors and 18 quadrangle-factors, namely a G-regular solution of HWP(48; 3,4; 5,18).
3.2 An octahedral solution of HWP(48; 3,4; 7,16)
Consider the seven cycles of Cycle(G) defined as follows.
Ci = (
C2 = ( C3 = ( C4 = (
C5 = (
C6 = ( C7 = (
-7(i + j), 2(1 - i + j + k)) 1 (-1 - i + j + k), 1 (1 - i - j - k)) 1 (-1 + i + j - k), 1 (-1 - i - j + k)) 7(-i + k), 1 (1 + i + j - k), -7(j + k))
72(i - ^ 72(1 - k), 72(1 + i)) 72(1 + k), -1 (1 + i + j + k), 72(1+ j))
-1 (1 + i + j + k), 1 (1 - i + j - k), 1 (1 - i - j + k))
We note that Sta6(G3) = V(C3) while all other Gj's have trivial stabilizer. Thus, by Lemma 2.2, one can check that Or6(Cj) is a £i-cycle decomposition of Cay[G : where li is the length of Cj and where the Qj's are the symmetric subsets of G listed below.
01	= {-7 (i+j ), 2 (1 - i+j + k), 7 (-j + k)}±1
02	= { 2 (-1 - i + j + k), 1 (1 - i - j - k), 2 (-1 - i + j - k)}
03	= {1 (-1 + i + j - k)}±1 = {72(-i + k) -72(1 - k) 72 (i + k), -72(j + k)}
O5 = {7(i - j), -j, 1 (1 - i + j - k), Tj(1 + i)}±1
±1
±1

{72 (1 + k), 72 (-1 + j) - 72 (1 + i), 72 (1 + j")}±1
O7 = {-1 (1 + i + j + k), -i, -k, J(1 - i - j + k)}
±1
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ArsMath. Contemp. 14 (2018) 1-14
One can see that the Q^s partition G \ {1, -1}. Thus, by Theorem 2.3 we can say that C := |JJ=1 OrbG (Gj) is a G-regular cycle-decomposition of K48 - I. Now set Fj = OrbSi(Gj) where
{K for i = 1,2; G for i = 3; L for 4 < i < 7.
By Remark 2.4, each Fj is a 2-factor of K48 — I with Sta6G(F) = Sj, hence Or6G(F) has length 3 or 1 or 4 according to whether i = 1, 2 or i = 3 or 4 < i < 7, respectively.
The cycles of Fj are triangles or quadrangles according to whether or not i < 3. Thus, recalling that C is a cycle-decomposition of K48 — I, we conclude that F := Ul=1 Or6G(Fj) is a G-regular 2-factorization of K48 — I with 7 triangle-factors and 16 quadrangle-factors, namely a G-regular solution of HWP(48; 3,4; 7,16).
3.3 An octahedral solution of HWP(48; 3,4; 9,14)
Consider the eight cycles of Gycle(G) defined as follows.
G1 =	, 7= (i + j), = (1 — i — j — k))	
G= = (	, —7= (1 — k), 7= (1+j))	
G3 = (	, 1 ( —1 — i + j + k), = (1 + i — j	
G4 = (	, 7= (—i + k) 7= (1 — ^ = ( —1	+ k)) ) — i + j — k))
G5 = (	, 7= (i — j), 1 ( —1 + i + j + k),	—7=(j +k))
Ge = (	, 7= (1 + ^ 7= (1 — i), 1 (1 — i	-j + k))
G7 = (	1, k, —1, —k)	
G8 = (	, j, —1, — j)	
We note that Siab(Gj) = V(Gj) for i = 7, 8 while all other Gj's have trivial stabilizer. By Lemma 2.2, one can check that Or6(Gj) is a ^-cycle decomposition of Cay[G : where £j is the length of Gj and where the Qj's are the symmetric subsets of G listed below.
Q = {7= (i + j), 2(1 — i — j — k), ^( —1 + i)}±1
Q = {—7= (1 — k), 7= (1 + j), 1 (—1 + i + j + k)}±1 Q = {=(—1 — i+j + k), 1 (1 + i — j + k), 2 (—1 — i — j + k)}±1 Q = {7= (—i + k), 1 (1 — i+j + k), 7= (i + k), 1 (—1 — i+j — k)}±1 q = {7= (i—j ) 7=(j —^ — 7= (1+j) — 7= (j+k)}±1 Q = {7= (1 + i), i, 7= (1 — k), = (1 — i—j + k)}±1 q = {k}±1 Q = {j}±1
Now note that the Q j's partition G\{ 1, — 1}. Thus, by Theorem 2.3 we can say that C := |J8=1 Or6(Gj) is a G-regular cycle-decomposition of K48 — I. Now set Fj = Or6S. (Gj)
S. Bonvicini and M. Buratti: Octahedral, dicyclic and special linear solutions of some
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where
K for 1 < i < 3; Si = < L for 4 < i < 6; G for i = 7,8.
By Remark 2.4, each Fj is a 2-factor of K48 - 1 with Sia&G(Fj) = Sj, hence Or6G(Fj) has length 3 or 4 or 1 according to whether 1 < i < 3 or 4 < i < 6 or i = 7, 8, respectively. The cycles of Fj are triangles or quadrangles according to whether or not i < 3. Thus, recalling that C is a cycle-decomposition of K48 - I, we conclude that F := U8=1 Or6G(Fj) is a G-regular 2-factorization of K48 -1 with 9 triangle-factors and 14 quadrangle-factors, namely a G-regular solution of HWP(48; 3,4; 9,14).
3.4 An octahedral solution of HWP(48; 3,4; 13,10)
Consider the nine cycles of Cycle(G) defined as follows.
Ci =	1,	-72(i + ^ -72(1+ j"))	
C2 =	(1,	1 (1 - i + j - k), -75(i + k))	
C3 =	(1,	7(-i+j), 7(1 - i -2 j - k))	
C4 =	(1,	7(-1 + i - j + k), 7J(i - k))	
C5 =	(1,	2(-1 - i + j + k), 2(-1 + i -	j - k))
C6 =	(1,	k, -1, -k)	
C7 =	(1,	j, -1, -j)	
C8 =	(1,	-2(1 + i + j + k), 2(-1 + i -	j' + k),
C9 =	(1,	-7(1 + k), -k, 7(-1 + i + j	- k))
72(
We note that Sta6(Cj) = V(Cj) for 5 < i < 7 while all other CVs have trivial G-stabilizer. Thus, by Lemma 2.2, one can check that Or6(Cj) is a ¿"¿-cycle decomposition of Cay[G : Qj] where ¿i is the length of Ci and where the ^'s are the symmetric subsets of G listed below.
" = {-7(i + j), -7^2 (1+ j), 2(1 + i + j - k)}±1
"2 = {2(1 - i + j - k), -7!(i + k), 7!(1 + i)}±7
"3 = {^(-i + j), 7(1 - i -2 j - k), ^ (j - k)}±7
"4 = {2(-1 + i - j + k), 7(i - k), -7(j + k)}±7 "5 = {2(-1 - i + j + k)}±7 "6 = {k}±7 "7 = {j}±7
"8 = {-7(1 + i + j + k), i, 7(-1 + i), 7(1 + j)}±7
"9 = {-7(1 + k), 7(1 - k), 7(1 - i + j + k), 2(-1 + i + j - k)}±7
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Ars Math. Contemp. 14 (2018) 97-116
Now note that the Q j's partition G\{ 1, -1}. Thus, by Theorem 2.3 we can say that C := Ui=1 Or6(Cj) is a G-regular cycle-decomposition of K48 - I. Now set Fj = OrbSi (Cj) where
{K for 1 < i < 4; G for 5 < i < 7; L for i = 8,9.
By Remark 2.4, each Fj is a 2-factor of K48 - I with Sta6G(Fj) = Sj, hence Or6G(Fj) has length 3 or 1 or 4 according to whether 1 < i < 4 or 5 < i < 7 or i = 8,9, respectively.
The cycles of F are triangles or quadrangles according to whether or not i < 5. Thus, recalling that C is a cycle-decomposition of K48 - I, we conclude that F := Ui=1 Or6G(F) is a G-regular 2-factorization of K48 - I with 13 triangle-factors and 10 quadrangle-factors, namely a G-regular solution of HWP(48; 3,4; 13,10).
3.5 An octahedral solution of HWP(48; 3,4; 15, 8)
Consider the seven cycles of Cycle(G) defined as follows.
C1 C2 C3
C4 =
(
C5 Ce = ( C7 = (
1 (-1 - i + j + k), -(i + k)) -75 (i + j) -75a + j))
2(-1 + i + j - k), 5(1 - i + j + k)) 7(1 + i + j + k), -1= (1+ j)) 5(1 - i+j - k), -(i - k))
-j k --72(1 - k))
—(i - j), 7(-1 - i + j - k), 5(-1 + i + j + k))
Here, every Cj has trivial stabilizer. Thus, by Lemma 2.2, one can check that Or6(Cj) is a ^-cycle decomposition of Cay[G : Q^ where £j is the length of Cj and where the Q^s are the symmetric subsets of G listed below.
Q1 = {5(-1 - i + j + k), - (i + k), - (-j + k)}±
= {--(i + j), --(1 + j), 5(1 + i + j - k)}
^ = {-75(i + j), -75 (1 1 j ), 5 (
l3 = {5 (-1 1 ° 1 j - 5 (1 - ° 1 j 1 5 1
l±1
Qa = {1 (-1 + i + j - k), 1 (1 - i + j + k), 5(-1 - i + j - k)}±1
Q4 = {1 (1 + i + j + k), -(1 + j), -(1 + i)}±1 Q5 = {1 (1 - i + j - k), -(i - k), -(j + k)}±1 Qe = {-j, +i, ^(1 - k), --(1 - k)}±1 Qr = {-(i - j), --(1 + i), +k, 1 (-1 + i + j + k)}±1
Ntw note that the Qj's partition G \ {1, -1}. Thus, by Theorem 2.3 we can say that C := Ur=1 Or6(Cj) is a G-regular cycle-decomposition of K48 - I. Set Fj = OrbSi (Cj) where
K for 1 < i < 5;
Sj =
L for i = 6, 7.
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By Remark 2.4, each Fi is a 2-factor of K48 — I with Sta6G(Fj) = Si, hence Or6G(Fj) has length 3 or 4 according to whether 1 < i < 5 or i = 6,7, respectively. The cycles of Fi are triangles or quadrangles according to whether or not i < 5. Thus, recalling that C is a cycle-decomposition of K48 — I, we conclude that F := |JJ=1 Or6G(Fi) is a G-regular 2-factorization of K48 — I with 15 triangle-factors and 8 quadrangle-factors, namely a G-regular solution of HWP(48; 3,4; 15,8).
3.6 An octahedral solution of HWP(48; 3,4; 17, 6)
Consider the ten cycles of Cycle(G) defined as follows.
Ci =
C2 = C3 = C4 = C5 = Ce = C7 = Cg = C9 =
-72(1 - k) -75 (i + k))
72V
-7(i + j), 2(-1 + i + j + k))
72
1 (1 + i - j - k), -7(1+ j)) 72 (-i+j) 72(-i + k))
1(1 - i + j - k), 7(1 - j)) i(-1 - i + j + k), i(-1 + i - j - k)) i(-1 + i +j - k), 2(-1 - i - j + k)) k, -1, -k) j, -1, -j)
C10 = (1, 7(1 + i), 7(1 - i), 2(1 - i - j + k))
We note that Stab(Ci) = V(Ci) for 6 < i < 9 while all other Ci's have trivial stabilizer. Thus, by Lemma 2.2, one can check that Orb(Ci) is a ¿¿-cycle decomposition of Cay[G : where li is the length of Ci and where the Qi's are the symmetric subsets of G listed below.
"1 = { — TTf (1 — k), — 72 (i + k), 1 ( —1 — i + j — k)}±1
n2
72v
{-72 (i + j), 2(-1 + i + j + k), 7!(-1 + i)l±1
n = {1 (1 + i - j - k), -7(1 + j), 7(j + k)}±1
^4 = { ^ (-i + j), ^ (-i + k), 1 (1 - i - j - k)}±1
^ = { 2 (1 - i + j - k), 7 (1 - j), 7 (j - k)}±1 n
^r = {2(-1 + i + j - k)}
n = {k}±1 n = {j}±1
{2 (-1 - i + j + k)}
±1
±1
n10 = {7(1 + i), i, 72 (1 - k), 1 (1 - i - j + k)}±1
Now note that the n4's partition G \ {1, -1}. Thus, by Lemma 2.2 we can say that
C := U^ Or6(Cj) is a G-regular cycle-decomposition of K48 - I. Set Fj = Or6S. (Cj)
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Ars Math. Contemp. 14 (2018) 97-116
where
{K for 1 < i < 5; G for 6 < i < 9; L for i = 10.
By Remark 2.4, each Fi is a 2-factor of K48 with Sta6G(Fj) = Si, hence Or6G(Fj) has length 3 or 1 or 4 according to whether 1 < i < 5 or 6 < i < 9 or i = 10, respectively. The cycles of Fi are triangles or quadrangles according to whether or not i < 7. Thus, recalling that C is a cycle-decomposition of K48 - I, we conclude that F := Ui=i Or6G(F) is a G-regular 2-factorization of K48 - / with 17 triangle-factors and 6 quadrangle-factors, namely a G-regular solution of HWP(48; 3,4; 17,6).
4 Dicyclic solutions of two Hamilton-Waterloo problems
In this section G will denote the dicyclic group of order 24 which is usually denoted by Q24. Thus G has the following presentation:
G = (a, b | a12 = 1, b2 = a6, b-1ab = a-1)
Note that the elements of G can be written in the form a1 b with 0 < i < 11 and j = 0,1. The group G has a unique involution which is a6 and we will need to consider the following subgroups of G:
•	H = (b) = {1, b, a6, a6b};
•	K = (a2) = {1, a2, a4, a6, a8, a10};
•	L = (a2b, a3) = {1, a3, a6, a9, a2b, a8b, a5b, a11b}.
4.1 A dicyclic solution of HWP(24; 3, 4; 7,4)
Consider the four cycles of Cycle(G) defined as follows.
C1 = (1, a3b, a5) C2 = (1, a10, a7b) C3 = (1, a4, a8) C4 = (1, b, a3b, a)
We note that the Stab(C3) = V(C3) while all other CVs have trivial stabilizer. Thus, by Lemma 2.2, one can check that Orb(Cj) is a ¿'¿-cycle decomposition of Cay[G : Qj] where ^ is the length of Q and where the Qi's are the symmetric subsets of G listed below.
= {a3b, a5, a2b}±1 Q2 = {a2, ab, a5b}±1 Q3 = {a4}±1 Q4 = {b, a3, a4b, a}±1
Now note that the Qi's partition G\{1, a6}. Thus, by Theorem 2.3 we can say that C := |J4=1 Orb(Ci) is a G-regular cycle-decomposition of K24 - I. Now set Fi = OrbSi (Ci)
S. Bonvicini and M. Buratti: Octahedral, dicyclic and special linear solutions of some
11
where
{L for i = 1, 2; G for i = 3; K for i = 4.
By Remark 2.4, each Fi is a 2-factor of K24 — I with StabG(Fi) = Si, hence OrbG(Fi) has length 3 or 1 or 4 according to whether i = 1, 2 or i = 3 or i = 4, respectively.
The cycles of Fi are triangles or quadrangles according to whether or not i < 3. Thus, recalling that C is a cycle-decomposition of K48 — I, we conclude that F := |J4=1 OrbG(Fi) is a G-regular 2-factorization of K24 — I with 7 triangle-factors and 4 quadrangle-factors, namely a G-regular solution of HWP(24; 3,4; 7,4).
4.2 A dicyclic solution of HWP(24; 3, 4; 9, 2)
Consider the four cycles of Cycle(G) defined as follows.
Ci =	(1,6, a6,a6b)
C2 =	(1, a46, a6, a1
C3 =	(l,a4,a76)
C4 =	(1, a36, a86)
C5 =	( 4 7 5) ^a , a , a J
We note that Stab(Ci) = V(Ci) for i = 1, 2 while all other Ci's have trivial stabilizer. By Lemma 2.2, one can check that Orb(Ci) is a ^-cycle decomposition of Cay[G : where li is the length of Ci and where the Qi's are the symmetric subsets of G listed below.
= {b}±1 ^2 = {a4b}±1
= {a4,ab, a5b}±1 = {a3b, a2b, a5}±1 ^5 = {a1,a2,a3}±1
Also here the Qi's partition G \ {1, a6}, hence C := |J5=1 OrbG(Ci) is a G-regular cycle-decomposition of K24 — I by Theorem 2.3. Now set:
F1 = OrbG(C1), F2 = OrbG(C2),
F3 = OrbL(Cs), F4 = OrbH(C4) U OrbH(C5).
By Remark 2.4, each Fi is a 2-factor of K24 — I and we have
StabG(F1) = StabG (F2) = G; StabG(F3) = L; StabG(F4) = H
so that the lengths of the G-orbits of F1, ..., F4 are 1, 1, 3 and 6, respectively. The cycles of Fi are triangles or quadrangles according to whether or not i > 3. Thus, recalling that C is a cycle-decomposition of K48 — I, we conclude that F := |J5=1 OrbG(Fi) is a G-regular 2-factorization of K24 — I with 9 triangle-factors and 2 quadrangle-factors, namely a G-regular solution of HWP(24; 3,4; 9, 2).
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Ars Math. Contemp. 14 (2018) 1-14
5 A special linear solution of HWP(24; 3,4; 5, 6)
In this section G will denote the 2-dimensional special linear group over Z3, usually denoted by SL2 (3), namely the group of 2 x 2 matrices with elements in Z3 and determinant one. The only involution of G is 2E where E is the identity matrix of G. The 2-Sylow subgroup Q of G, isomorphic to the group of quaternions, is the following:
Q =
1 0 0 1
2 0 0 2
11 12
22 21
02 10
01 20
12 22
21 11
We will also need to consider the subgroup H of G of order 6 generated by the matrix 0 Jl. Hence we have:
H
10 01
01 21
21 20
20 02
02 12
12 10
The use of the special linear group G was crucial in [5] to get a Steiner triple system of any order v = 144n + 25 with an automorphism group acting sharply transitively an all but one point. Here G will be used to get a G-regular solution of the last Hamilton-Waterloo problem left open in [11].
Consider the six cycles of Cycle(G) defined as follows.
Ci = C2 = C3 = C4 = C5 = Ce =
1	0		2	0		1	2	\		
0	1		2	2		1	0	)		
1	0		0	2		2	1	)		
0	1		1	2		2	0	)		
1	0		0	1		2	2	)		
0	1		2	2		1	0	)		
1	0		0	1		2	0		0	2
0	1		2	0		0	2		1	0
1	0		1	1		2	0		2	2
0	1		1	2		0	2		2	1
1	0		2	1		2	1		1	1
0	1		1	1		0	2		0	1
Here the stabilizer of Ci is trivial for i = 1, 6 while it coincides with V(Ci) for 2 < i < 5. By Lemma 2.2, one can check that Orb(Ci) is a ¿'¿-cycle decomposition of Cay[G : Q^ where ¿i is the length of Cj and where the Qi's are the symmetric subsets of G listed below.
±1
fi =

20 22
1	2		0	2
1	0		1	1

fie =
±1
±1
fia
fi.5
01
22
±1
±1
10 21
22 02
±1
S. Bonvicini and M. Buratti: Octahedral, dicyclic and special linear solutions of some
13
Once again we see that the Qj's partition G \ {E, 2E}, therefore C := |J6=1 Orb(Ci) is a G-regular cycle-decomposition of K24 - I. Now set Fi = OrbSi (Ci) with
(Q for i = 1;
Si = < G for 2 < i < 5; [h for i = 6.
By Remark 2.4, each Fi is a 2-factor of K24 - I and we have StabG(Fi) = Si so that the lengths of the G-orbits of F1,..., F6 are 3, 1, 1, 1, 1 and 4, respectively.
The cycles of Fi have length 3 or 4 according to whether or not i < 3. Thus, recalling that C is a cycle-decomposition of K24 - I, we conclude that F := |J6=1 OrbG(Fi) is a G-regular 2-factorization of K24 - I with 5 triangle-factors and 6 quadrangle-factors, namely a G-regular solution of HWP(24; 3,4; 5,6).
References
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