Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 6 (2013) 147–154
Quasi m-Cayley circulants
Ademir Hujdurović ∗
University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia
University of Primorska, IAM, Muzejski trg 2, 6000 Koper, Slovenia
Received 30 October 2011, accepted 4 February 2012, published online 15 June 2012
Abstract
A graph Γ is called a quasi m-Cayley graph on a group G if there exists a vertex
∞ ∈ V (Γ) and a subgroup G of the vertex stabilizer Aut(Γ)∞ of the vertex∞ in the full
automorphism group Aut(Γ) of Γ, such that G acts semiregularly on V (Γ)\{∞} with m
orbits. If the vertex∞ is adjacent to only one orbit of G on V (Γ)\{∞}, then Γ is called
a strongly quasi m-Cayley graph on G. In this paper complete classifications of quasi
2-Cayley, quasi 3-Cayley and strongly quasi 4-Cayley connected circulants are given.
Keywords: Arc-transitive, circulant, quasi m-Cayley graph.
Math. Subj. Class.: 05C15, 05C10
1 Introduction
Throughout this paper graphs are assumed to be finite, simple, connected and undi-
rected, and groups are finite. Given a graph Γ we let V (Γ), E(Γ), A(Γ) and Aut(Γ) be the
set of its vertices, edges, arcs and the automorphism group of Γ, respectively. A graph Γ
is said to be vertex-transitive, edge-transitive, and arc-transitive if its automorphism group
acts transitively on V (Γ), E(Γ) and A(Γ), respectively.
Let G be a finite group with identity element 1, and let S ⊂ G\{1} be such that
S−1 = S. We define the Cayley graph Cay(G,S) on the group G with respect to the
connection set S to be the graph with vertex set G, in which two vertices x, y ∈ G are
adjacent if and only if x−1y ∈ S. A circulant of order n is a Cayley graph on a cyclic
group of order n.
In this paper we consider quasi-semiregular actions on graphs, a natural generalization
of semiregular actions on graphs, which have been an active topic of research in the last
decades (see, for example, [1, 2, 3, 4, 5, 8, 9, 11]). Following [7] we say that a groupG acts
∗Supported in part by “Agencija za raziskovalno dejavnost Republike Slovenije”, research program P1-0285
and “Mladi raziskovalec” research program.
E-mail address: ademir.hujdurovic@upr.si (Ademir Hujdurović)
Copyright c© 2013 DMFA Slovenije
148 Ars Math. Contemp. 6 (2013) 147–154
quasi-semiregularly on a set X if there exists an element∞ in X such that G fixes∞, and
the stabilizer Gx of any element x ∈ X\{∞} is trivial. The element∞ is called the point
at infinity. A graph Γ is called quasim-Cayley onG if the groupG acts quasi-semiregularly
on V (Γ) with m orbits on V (Γ)\{∞}. If G is cyclic and m = 1 (respectively, m = 2,
m = 3 and m = 4) then Γ is said to be quasi circulant (respectively, quasi bicirculant,
quasi tricirculant and quasi tetracirculant). In addition, if the point at infinity∞ is adjacent
with only one orbit of G∞ then we say that Γ is a strongly quasi m-Cayley graph on G.
Quasi m-Cayley graphs were first defined in 2011 by Kutnar, Malnič, Martinez and
Marušič [7], who showed which strongly quasi m−Cayley graphs are strongly regular
graphs.
In this paper, we consider which circulants are also quasi m-Cayley graphs. Our main
results are stated in the following three theorems.
Theorem 1.1. Let Γ be a quasi 2-Cayley graph of order n which is also a connected circu-
lant. Then either Γ is isomorphic to the complete graph Kn, or n ≡ 1 (mod 4) is a prime
and Γ is isomorphic to the Paley graph P (n). Moreover, Γ is a quasi bicirculant.
Theorem 1.2. Let Γ be a connected circulant. Then Γ is also a quasi 3-Cayley graph if and
only if either Γ = Kn, or replacing Γ with its complement if necessary, Γ ∼= Cay(Zn, S),
where S is the set of all non-zero cubes in Zn, and n is a prime such that n ≡ 1 (mod 3).
Moreover, Γ is a quasi tricirculant.
Theorem 1.3. Let Γ be a connected circulant. Then Γ is a strongly quasi 4-Cayley graph
on a group G if and only if Γ ∼= C9 or Γ ∼= Cay(Zn, S), where S is the set of all fourth
powers in Zn\{0}, and n is a prime such that n ≡ 1 (mod 4). Moreover, Γ is a quasi
tetracirculant.
The paper is organized as follows. In Section 2 we recall the classification of connected
arc-transitive circulants. In Section 3 we prove Theorem 1.1 and in Section 4 we prove
Theorems 1.2 and 1.3.
2 Arc-transitive circulants
We begin this section with the following lemma:
Lemma 2.1. Let Γ be a connected vertex-transitive strongly quasi m−Cayley graph. Then
Γ is arc-transitive.
Proof. Since Γ is vertex transitive, it is sufficient to prove that there exists a vertex v such
that the stabilizer Aut(Γ)v acts transitively on the neighborhood of v. It is obvious that if
we choose the point at infinity for v, this condition is satisfied.
The previous lemma implies that we can somehow restrict our study to the connected
arc-transitive circulants, therefore it is important to understand the structure of such graphs.
To state the classification of connected arc-transitive circulants, which has been ob-
tained independently by Kovács [6] and Li [10], we need to recall certain graph products
and the concept of normal Cayley graphs.
The wreath (lexicographic) product Σ[Γ] of a graph Γ by a graph Σ is the graph with
vertex set V (Σ) × V (Γ) such that {(u1, u2), (v1, v2)} is an edge if and only if either
{u1, v1} ∈ E(Σ), or u1 = v1 and {u2, v2} ∈ E(Γ). For a positive integer b and a graph
A. Hujdurović: Quasi m-Cayley circulants 149
Σ, denote by bΣ the graph consisting of b vertex-disjoint copies of the graph Σ. The graph
Σ[Kb]−bΣ is called the deleted wreath (deleted lexicographic) product of Σ andKb, where
Kb = bK1.
Let Γ = Cay(G,S) be a Cayley graph on a group G. Denote by Aut(G,S) the set of
all automorphisms of G which fix S setwise, that is,
Aut(G,S) = {σ ∈ Aut(G)|Sσ = S}.
It is easy to check that Aut(G,S) is a subgroup of Aut(Γ) and that it is contained in the
stabilizer of the identity element 1 ∈ G. It follows from the definition of Cayley graph that
the left regular representation GL of G induces a regular subgroup of Aut(Γ). Following
Xu [12], Γ = Cay(G,S) is called a normal Cayley graph if GL is normal in Aut(Γ), that
is, if Aut(G,S) coincides with the vertex stabilizer 1 ∈ G. Moreover, if Γ is a normal
Cayley graph, then Aut(Γ) = GL o Aut(G,S).
Proposition 2.1. [6, 10] Let Γ be a connected arc-transitive circulant of order n. Then one
of the following holds:
(i) Γ ∼= Kn;
(ii) Γ = Σ[Kd], where n = md, m, d > 1 and Σ is a connected arc-transitive circulant
of order m;
(iii) Γ = Σ[Kd] − dΣ, where n = md, d > 3, gcd(d,m) = 1 and Σ is a connected
arc-transitive circulant of order m;
(iv) Γ is a normal circulant.
In Section 3 and 4 two lemmas (that show that arc-transitive circulants described in
Proposition 2.1(ii) and (iii) are not strongly quasi k-Cayley graphs) will be needed.
Lemma 2.2. Let Γ be an arc-transitive circulant, described in Proposition 2.1(ii). Then Γ
is not a strongly quasi k-Cayley graph for any k ∈ N.
Proof. We have Γ = Σ[Kd], where n = md, m, d > 1 and Σ is a connected arc-transitive
circulant of order m. Suppose that Γ is a strongly quasi k-Cayley graph on a group G.
Then val(Γ) = (n − 1)/k = (md − 1)/k. On the other hand, since Γ = Σ[Kd], we have
val(Γ) = val(Σ) ·d. These two facts combined together imply that d(m−k ·val(Σ)) = 1,
and so d = 1, a contradiction.
Lemma 2.3. Let Γ be an arc-transitive circulant, described in Proposition 2.1(iii). Then Γ
is not a strongly quasi k-Cayley graph for any k ∈ N.
Proof. We have Γ = Σ[Kd] − dΣ, where n = md, d > 3, gcd(d,m) = 1, and Σ is an
arc-transitive circulant of order m. Suppose that Γ is also a strongly quasi k-Cayley graph
on a group G. By [10, Theorem 1.1] the m copies of the graph Kd form an imprimitivity
block system B for Aut(Γ). Clearly the blockB ∈ B containing the point at infinity, that is,
the trivial orbit of G, is fixed by G. This implies that |G| divides d− 1. On the other hand,
since the valency of Γ is |G|, we have |G| ≥ d − 1. Combining these results we obtain
|G| = d−1. Thus, connectedness of Γ implies that m = 2. However, then there are 2d−1
vertices in Γ different from the point at infinity, and they cannot be divided into k orbits of
size d−1 for any natural number k. Therefore, there are no strongly quasi k-Cayley graphs
amongst the graphs from Proposition 2.1(iii) for any natural number k ≥ 1.
150 Ars Math. Contemp. 6 (2013) 147–154
Lemma 2.4. Let Γ be an arc-transitive circulant, described in Proposition 2.1(iv). If Γ is
also a strongly quasi m-Cayley graph on a group G, then the order of Γ has at most m+ 1
divisors.
Proof. Let Γ = Cay(Zn, S) be a normal circulant. Let A = Aut(Γ). Since Γ is a normal
Cayley graph, A ∼= Zn o Aut(Zn, S). We may, without loss of generality, assume that the
point at infinity corresponds to the vertex 0 ∈ Zn, and so G ≤ Aut(Zn, S) ≤ Aut(Zn) ∼=
Z∗n. Therefore, G . Z∗n. Since G has m orbits on Zn\{0}, then Aut(Zn) has at most m
orbits on Zn\{0}, and at most m+ 1 orbits on Zn. Elements in the same orbit of Aut(Zn)
are clearly of the same order in Zn. There exist an element in Zn of order d, if and only
if d divides n. Therefore the number of divisors of n, denoted by τ(n), is not greater than
m+ 1, i.e. τ(n) ≤ m+ 1.
3 Quasi 2-Cayley graphs
In this section the connected circulants are considered. In particular, connected circu-
lants that are also quasi 2-Cayley graphs are classified (see Theorem 1.1). If a graph Γ of
order n is a quasi 2-Cayley graph on a group G, which is not a strongly quasi 2-Cayley
graph, then it is isomorphic to the complete graph Kn. Namely, in such a graph, the point
at infinity ∞ is adjacent to both nontrivial orbits of G, and thus it is adjacent to all the
vertices different from∞. Consequently, we can conclude that Γ has valency |V (Γ)| − 1,
and so Γ is a complete graph. In order to classify all connected circulants that are also
quasi 2-Cayley graphs it therefore suffices to characterize strongly quasi 2-Cayley graphs
that are also connected circulants, we do this in Theorem 3.1.
Theorem 3.1. Let Γ be a connected circulant. Then Γ is also a strongly quasi 2-Cayley
graph if and only if Γ is isomorphic to the Paley graph P (p), where p is a prime such that
p ≡ 1 (mod 4). Moreover, Γ is a quasi bicirculant.
Proof. Let Γ be the Paley graph P (p), where p is a prime, such that p ≡ 1 (mod 4). It
is well known that the Paley graphs are connected arc-transitive circulants, and, as was
observed in [7], they are also strongly quasi 2-Cayley graphs.
Conversely, let Γ be a connected circulant Cay(Zn, S) of order n not isomorphic to
the complete graph Kn, which is also a strongly quasi 2-Cayley graph on a group G. Then
|G| = (n−1)/2 and Γ is of valency (n−1)/2. Lemma 2.1 tells us that Γ is an arc-transitive
graph, and moreover Proposition 2.1, Lemma 2.2 and Lemma 2.3 combined together imply
that Γ is a normal circulant. The theorem now follows from the three claims below.
CLAIM 1: n is an odd prime.
It is obvious that n must be odd, since 2 divides n − 1. By Lemma 2.4 we have that
τ(n) ≤ 3. Thus we have the following two possibilities for n:
• n = p, where p is a prime;
• n = p2, where p is a prime.
Suppose that the latter case hold. Let A = Aut(Γ). Since Γ is a normal Cayley graph, we
have A ∼= Zn o Aut(Zn, S). We may, without loss of generality, assume that the point at
infinity corresponds to the vertex 0 ∈ Zn, and so G ≤ Aut(Zn, S) ≤ Aut(Zn) ∼= Z∗n.
Therefore, Z∗n contains a subgroup G of order (n − 1)/2. Since |Z∗n| ≤ n − 1 and |G|
A. Hujdurović: Quasi m-Cayley circulants 151
divides |Z∗n| we obtain that |Z∗n| = n − 1 or (n − 1)/2. Since, by assumption, n is not a
prime, we have |Z∗n| = (n− 1)/2. This gives in the following equation
p2 − 1
2
= p(p− 1)
which has the unique solution p = 1, a contradiction.
CLAIM 2: n ≡ 1 (mod 4).
Since S = −S, and no element in Zn can be its own inverse, we have that the number of
elements in S is even, and since |S| = n−12 , we have n ≡ 1 (mod 4).
CLAIM 3: Γ is isomorphic to the Paley graph P (n).
By Claim 1, n is a prime. Therefore the group Z∗n is cyclic, and thus since G is a subgroup
of Z∗n, G is cyclic as well. By [6, Remark 2], we have Aut(Γ) = {g 7→ gσ + h | σ ∈
K,h ∈ Zn}, for a suitable groupK < Aut(Zn), and S is the orbit underK of a generating
element of Zn, that is, S = OrbK(g) for some generating element g of Zn. Now we have
that Aut(Γ)0 = {g 7→ gσ +h | σ ∈ K,h ∈ Zn : 0σ +h = 0} = {g 7→ gσ | σ ∈ K} ∼= K.
So we see that G . K. Since S = OrbK(g) . OrbG(g), and |S| = |OrbG(g)| we have
that S ∼= OrbG(g), which gives us that S ∼= G (taking g = 1). Now, since G is the index
2 subgroup of the cyclic group Z∗n, G is of the form G = 〈x2〉 where x generates Z∗n.
Therefore G consists of all squares in Z∗n and S ∼= G, implying that Γ is isomorphic to the
Paley graph P (n) as claimed.
It is obvious that G must be cyclic, so the graph Γ is in fact a quasi bicirculant.
Proof of Theorem 1.1: It follows from Theorem 3.1 and the paragraph preceding it. 
In general, if Γ is a vertex transitive quasi 2-Cayley graph on a groupG, not isomorphic
to the complete graph, then it is a strongly regular graph of a rank 3 group. Namely, the
orbits of G are contained in the orbits of the stabilizer of the Aut(Γ)∞ and since there are
just two nontrivial orbits of G, then there are exactly two nontrivial orbits of the Aut(Γ)∞
which in fact must coincide with the orbits ofG. Therefore Aut(Γ) must be a rank 3-group,
and the graphs of the rank 3 groups are strongly regular graphs.
4 Quasi 3-Cayley and 4-Cayley graphs
In this section we will deal with the question which connected circulants are also quasi
3-Cayley graph or strongly quasi 4-Cayley graphs. We first consider the case of strongly
quasi 3-Cayley graphs.
Theorem 4.1. Let Γ be a connected circulant. Then Γ is also a strongly quasi 3-Cayley
if and only if Γ ∼= Cay(Zn, S) where S is the set of all non-zero cubes in Zn, and n is a
prime such that n ≡ 1 (mod 3). Moreover, Γ is a quasi tricirculant.
Proof. Let Γ = Cay(Zp, S) where p ≡ 1 (mod 3) is a prime and S is the set of all non-
zero cubes in Zp. Since p is a prime, it is well known that Aut(Zp) ∼= Z∗p is a cyclic group
of order p−1. Let G = 〈a3〉, where a is a generating element of Z∗p. Then G consists of all
non-zero cubes in Zp, and |G| = p−13 . The action of G on Zp defined by x
g = g · x gives
G as the subgroup of Aut(Γ). The group G acts quasi-semiregularly on Zp with 0 ∈ Zp
152 Ars Math. Contemp. 6 (2013) 147–154
as the point at infinity. Namely, it is easy to check that G0 = G, and that the stabilizer
of any element x ∈ Zp\{0} is trivial. Since |G| = p−13 , it follows that G has 3 orbits on
Zp\{0}, and therefore Γ is a quasi 3-Cayley graph. Since one of the orbits of G is the set
S, the point at infinity is adjacent to only one orbit of G, so Γ is in fact a strongly quasi
3-Cayley graph. By the construction Γ is an arc-transitive circulant since G ≤ Aut(Γ)0
acts transitively on the set of vertices adjacent to the vertex 0.
Conversely, let Γ be a connected circulant of order n, which is also a strongly quasi
3-Cayley graph on a group G. Then |G| = n−13 . From Lemma 2.1 we have that Γ is arc-
transitive, and therefore Proposition 2.1, Lemma 2.2 and Lemma 2.3 combined together
imply that Γ is a normal circulant. Therefore, we can assume that Γ = Cay(Zn, S), and
that G ≤ Aut(Zn, S) ≤ Aut(Zn), implying that n−13 |ϕ(n), where ϕ(n) is the Euler totient
function.
CLAIM 1: n is a prime number.
Let
n = pk11 · p
k2
2 · · · p
kt
t ,
be a canonic factorization of a positive integer n. From Lemma 2.4, we have τ(n) ≤ 4.
Now we can calculate
τ(n) = (k1 + 1)(k2 + 1) · · · (kt + 1).
We have the following possibilities for n:
• n = p,
• n = p2;
• n = p3;
• n = pq,
where p and q are different primes.
If n = p2, then the only solution of n−13 |ϕ(n) is p = 2 and n = 4. However, if n = 4,
the graph Γ is of valency 1, so it is not a connected graph.
If n = p3, then there is no solution of the above equation.
If n is a product of two different primes, then we have |Z∗n| = (n−1)/3 or 2(n−1)/3.
In the first case Z∗n ∼= G, so Z∗n acts semiregularly on Zn\{0}, and it is not difficult to see
that this is not the case for n = pq. If |Z∗n| = 2(n − 1)/3, then we obtain the following
equation
(p− 1)(q − 1) = 2(pq − 1)
3
.
The only solutions in natural numbers of the above equation are (p, q) ∈ {(4, 7), (5, 5),
(7, 4)}, so there are no two different primes p, q satisfying the given equation.
Having in mind all the written above, we conclude that n is a prime.
CLAIM 2: Γ is isomorphic to the Cayley Graph Cay(Zn, S), where S is set of all non zero
cubes in Zn, and n is a prime such that n ≡ 1 (mod 3).
Similarly as in the previous section, it can be shown that G ∼= S. Since G is an index 3
subgroup of Z∗n, we have G = 〈x3〉, where x is a generating element of Z∗n. It follows that
G consists of all cubes in Z∗n, so Γ is isomorphic to Cay(Zn, S), where S is the set of all
non zero cubes in Zn and n ≡ 1 (mod 3) is a prime. It is obvious from the mentioned
above, that the group G must be cyclic, therefore, Γ is in fact a quasi tricirculant.
A. Hujdurović: Quasi m-Cayley circulants 153
Proof of the Theorem 1.2: Let Γ be a connected circulant of order n, which is also a quasi
3-Cayley on a group G. The point at infinity is adjacent to all three nontrivial orbits of G,
if and only if Γ is isomorphic to Kn. If the point at infinity is adjacent to just one nontrivial
orbit of G, then Γ is a strongly quasi 3-Cayley graph, therefore, Theorem 4.1 gives us the
desired result. If the point at infinity is adjacent to two nontrivial orbits of G, then we
consider the complement Σ = Γ of the graph Γ. The graph Σ is a quasi 3-Cayley graph
on G, and actually it is a strongly quasi 3-Cayley graph on G. Since Σ is the complement
of a circulant it is also a circulant. Suppose that Σ is not connected. Then, since it is
vertex-transitive, it is the disjoint union of some isomorphic graphs. The point at infinity is
adjacent to one orbit of G, so the connected components of Σ must have at least 1 + n−13
points. Therefore n = k · n1, where k is the number of connected components, and n1 is
the number of points in each of the components. We have noticed that n1 ≥ 1 + n−13 , thus
k ≤ 2. If k = 1 then Σ is connected. Suppose that k = 2. Then there are two connected
components of Γ, say Γ1 and Γ2, each containing n/2 points. Suppose that∞ ∈ Γ1. Let
∆1,∆2 and ∆3 be a nontrivial orbits of G, and let the point at infinity be adjacent to ∆1.
Then ∆1 ⊂ Γ1. Since Γ1 and Γ2 have the same size, it means that at least one of ∆2 and ∆3
have points both in Γ1 and Γ2. Suppose that u, v ∈ ∆2, such that u ∈ Γ1 and v ∈ Γ2. Since
u and v are in the same orbit of G then there exist g ∈ G which maps u to v. However, g
fixes∞, and consequently g fixes Γ1, a contradiction.
Having in mind all the written above, we see that Σ is a connected circulant, which is
also a strongly quasi 3-Cayley graph. Therefore we have the desired result. 
We will continue this section with the proof of Theorem 1.3.
Proof of Theorem 1.3: Let Γ = C9. Then Γ ∼= Cay(Z9, {±1}). Then the group G =
{1,−1} ⊂ Z∗9 acts quasi semiregularly on Z9 with 0 as the point at infinity.
Let Γ ∼= Cay(Zp, S), where S is the set of all fourth powers in Zp\{0}, and p is a
prime such that p ≡ 1 (mod 4). Define G = 〈a4〉, where a is some generating element
of Z∗p, which is cyclic in this case. We have that G acts quasi-semiregularly on Z∗p with 0
as the point at infinity. Since |G| = p−14 , it follows that G has 4 orbits on Zp\{0}, and
therefore Γ is a quasi 4-Cayley graph. It is also easy to see that 0 is adjacent to only one
orbit of G on Zp\{0}, therefore Γ is a strongly quasi 4-Cayley graph. By the construction,
Γ is a connected arc-transitive circulant.
Conversely, let Γ be a connected circulant of order n which is also a strongly quasi
4-Cayley graph on a group G. Then |G| = (n − 1)/4. Using Lemma 2.1 we have that
Γ is arc-transitive, and so Proposition 2.1, Lemma 2.2 and Lemma 2.3 combined together
imply that Γ is a normal circulant. Therefore, we can assume that Γ = Cay(Zn, S), and
that G ≤ Aut(Zn, S) ≤ Aut(Zn), implying that
n− 1
4
|ϕ(n). (4.1)
Using Lemma 2.4 we obtain τ(n) ≤ 5. So we have the following possibilities:
• n = p,
• n = p2,
• n = p3,
• n = p4,
• n = pq,
154 Ars Math. Contemp. 6 (2013) 147–154
where p and q are different primes.
If n = p2, then the only solution of (4.1) is n = 9. In this case, the valency of Γ is
(9− 1)/4 = 2, so Γ ∼= C9.
In the cases when n = p3, and n = p4 there is no prime satisfying (4.1).
When n = pq, we have that (p − 1)(q − 1) = α · (pq − 1)/4, where α ∈ {1, 2, 3}.
If α = 1, then we have Z∗n = G, so Z∗n must act semiregularly on Zn\{0}, which is not
the case. If α = 2, then there are no two different primes satisfying (p − 1)(q − 1) =
(pq − 1)/2, and finally, when α = 3, we have that n = 5 · 13 is the only possibil-
ity. In this case, Γ is a connected arc-transitive circulant on 65 vertices, which has va-
lency 16. Since G is an index 3 subgroup of Z∗65 ∼= Z4 × Z12, then we can calcu-
late G ∼= {±1,±8,±12,±14,±18,±21,±27,±31}, and we can see that G does not act
semiregularly on Z65\{0}. Namely, the non identity element 21 ∈ G fixes the point
13 ∈ Z65\{0}.
Assume now that n is a prime. Similarly as in the proof of Theorem 3.1, we obtain
G ∼= S, and therefore, since G is an index 4 subgroup of Z∗n, we have G = 〈x4〉, where x
is some generating element of Z∗n. Therefore, Γ ∼= Cay(Zn, S), where S is the set of all
fourth powers in Zn\{0}.
From the mentioned above, it is clear that G is a cyclic group, so Γ is in fact a quasi
tetracirculant. 
References
[1] E. Dobson, A. Malnič, D. Marušič and L. A. Nowitz, Minimal normal subgroups of transitive
permutation groups of square-free degree, Discrete Math. 307 (2007), 373–385.
[2] E. Dobson, A. Malnič, D. Marušič and L. A. Nowitz, Semiregular automorphisms of vertex-
transitive graphs of certain valencies. J. Combin. Theory Ser. B 97 (2007), 371–380.
[3] M. Giudici, Quasiprimitive groups with no fixed point free elements of prime order, J. London
Math. Soc. 67 (2003), 73–84.
[4] M. Giudici, New constructions of groups without semiregular subgroups, Comm. Algebra 35
(2007), 2719–2730.
[5] M. Giudici and J. Xu, All vertex-transitive locally-quasiprimitive graphs have a semiregular
automorphism, J. Algebr. Combin. 25 (2007), 217–232.
[6] I. Kovács, Classifying arc-transitive circulants, J. Algebr. Combin. 20 (2004), 353–358.
[7] K. Kutnar, A. Malnič, L. Martinez and D. Marušič, Quasi m-Cayley strongly regular graphs,
manuscript.
[8] K. Kutnar and D. Marušič, Recent trends and future directions in vertex-transitive graphs, Ars
Math Contemp. 1 (2008), 112–125.
[9] K. Kutnar and P. Šparl, Distance-transitive graphs admit semiregular automorphisms, European
J. Combin. 31 (2010), 25–28.
[10] C. H. Li, Permutation groups with a cyclic regular subgroup and arc-transitive circulants, J.
Algebr. Combin. 21 (2005), 131–136.
[11] D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981), 69–81.
[12] M. Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182
(1998), 309–320.