In this paper, we prove that cyclic hamiltonian cycle systems of the complete graph minus a 1-factor, K n − I, exist if and only if n ≡ 2, 4(mod 8) and n = 2p α with p prime and α ≥ 1.
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. The work in this paper makes… (More)
A circulant (di)graph is a (di)graph on n vertices that admits a cyclic automorphism of order n. This paper provides a survey of the work that has been done on finding the automorphism groups of circulant (di)graphs, including the generalisation in which the edges of the (di)graph have been assigned colours that are invariant under the aforementioned cyclic… (More)
A balanced graph is a bipartite graph with no induced circuit of length 2 (mod 4). These graphs arise in linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove that there is no cubic balanced planar graph.… (More)
It has been shown that there is a Hamilton cycle in every connected Cayley graph on any group G whose commutator subgroup is cyclic of prime-power order. This note considers connected, vertex-transitive graphs X of order at least 3, such that the automorphism group of X contains a vertex-transitive subgroup G whose commutator subgroup is cyclic of… (More)
Let S be a subset of the units in n. Let Γ be a circulant graph of order n (a Cayley graph of n) such that if ij ∈ E(Γ), then i − j (mod n) ∈ S. Toida conjectured that if Γ is another circulant graph of order n, then Γ and Γ are isomorphic if and only if they are isomorphic by a group automorphism of n. In this paper, we prove that Toida's conjecture is… (More)
A circulant is a Cayley graph of a cyclic group. Arc-transitive circulants of square-free order are classiied. It is shown that an arc-transitive circulant ? of square-free order n is one of the following: the lexicographic product K b ], or the deleted lexicographic K b ] ? b, where n = bm and is an arc-transitive circulant, or ? is a normal circulant,… (More)
We strengthen a classical result of Sabidussi giving a necessary and sufficient condition on two graphs, X and Y , for the automorphsim group of the wreath product of the graphs, Aut(X Y) to be the wreath product of the auto-morphism groups Aut(X) Aut(Y). We also generalize this to arrive at a similar condition on color digraphs. The main purpose of this… (More)