Joy Morris

Automorphism groups with cyclic commutator subgroup and Hamilton cycles

Abstract It has been shown that there is a Hamilton cycle in every connected Cayley graph on each group G whose commutator subgroup is cyclic of prime-power order. This paper considers connected, vertex-transitive graphs X of order at least 3 where the automorphism group of X contains a transitive subgroup G whose commutator subgroup is cyclic of prime-power order. We show that of these graphs, only the Petersen graph is not Hamiltonian.

ON THE MAXIMUM ORDERS OF ELEMENTS OF FINITE ALMOST SIMPLE GROUPS AND PRIMITIVE PERMUTATION GROUPS

We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many excep- tions, the maximum element order is at most m(T). Moreover, apart from an explicit list of groups, the bound can be reduced to m(T)/4. These results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n/4. We note again that this result gives upper bounds for meo(Aut(T)) in terms of m(T), and for meo(G) in terms of m(G) (since m(T) ≤ m(G)). Moreover equality in the up- per bound meo(Aut(T)) ≤ m(T) holds when T = PSLd(q) for all but two pairs (d,q), see Table 3 and Theorem 2.16. (Theorem 2.16 and Table 3 provide good estimates for

Hamiltonian cycles in Cayley graphs whose order has few prime factors

We prove that if Cay( G ; S ) is a connected Cayley graph with n vertices, and the prime factorization of n is very small, then Cay( G ; S ) has a hamiltonian cycle. More precisely, if p , q , and r are distinct primes, then n can be of the form kp with 24 ≠ k < 32, or of the form kpq with k ≤ 5, or of the form pqr , or of the form kp 2 with k ≤ 4, or of the form kp 3 with k ≤ 2.

On automorphism groups of circulant digraphs of square-free order

We show that the full automorphism group of a circulant digraph of square-free order is either the intersection of two 2-closed groups, each of which is the wreath product of 2-closed groups of smaller degree, or contains a transitive normal subgroup which is the direct product of two 2-closed groups of smaller degree.

Isomorphic Cayley graphs on nonisomorphic groups

It was conjectured in [Wang, to appear in The Australasian Journal of Combinatorics] that, for each integer k ≥ 2, there exists M(k) such that if G= (V1, V2, E) is a bipartite graph with |V1| = |V2| = n ≥ M(k) and d(x) + d(y) ≥ n + k for each pair of nonadjacent vertices x and y of G with x e V1 and y e V2, then, for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck in G such that ei e E(Ci for all i e {1, …, k} and V(C1 ⊎ ··· ∪ Ck) = V(G). This conjecture is also verified for k = 2, 3 in [Wang, to appear; Wang, manuscript]. In this article, we prove this conjecture to be true if n ≥ 3k, i.e., M(k) ≤ 3k. We will also show that, if n ≥ 3k, then for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck of length at most 6 in G such that ei e E(Ci) for all i e {1, …, k}.

Automorphisms of Cayley graphs on generalised dicyclic groups

A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs: abelian groups and generalised dicyclic groups (Babai and Godsil, 1982). Indeed, any Cayley graph on such a group admits specific additional graph automorphisms that depend only on the group. Recently, Dobson and the last two authors showed that almost all Cayley graphs on abelian groups admit no automorphisms other than these obvious necessary ones (Dobson et?al., in press). In this paper, we prove the analogous result for Cayley graphs on the remaining family of exceptional groups: generalised dicyclic groups.

Generalised quadrangles with a group of automorphisms acting primitively on points and lines

We show that if G is a group of automorphisms of a thick finite generalised quadrangle Q acting primitively on both the points and lines of Q, then G is almost simple. Moreover, if G is also flag-transitive then G is of Lie type.

Strongly regular edge-transitive graphs

In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parameters of the graphs in this family that reduce to complete graphs.

Classifying Arc-Transitive Circulants of Square-Free Order

AbstractA circulant is a Cayley graph of a cyclic group. Arc-transitive circulants of square-free order are classified. It is shown that an arc-transitive circulant Γ of square-free order n is one of the following: the lexicographic product