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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 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)

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)

The issue of when two Cayley digraphs on different abelian groups of prime power order can be isomorphic is examined. This had previously been determined by Anne Joseph for squares of primes; her results are extended. 1 Preliminaries We begin with some essential definitions. For many of the results in this paper, the lemmata and proofs used are direct… (More)

A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. God-sil has shown that there are only two infinite families of finite groups that do not admit GRRs: abelian groups and generalised dicyclic groups [4]. Indeed, any Cayley graph on such a group admits specific additional graph… (More)

- Dave Witte, Morris, Joy Morris, Gabriel Verret
- 2016

Let S be a finite generating set of a torsion-free, nilpo-tent group G. We show that every automorphism of the Cayley graph Cay(G; S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G1; S1) and Cay(G2; S2) are connected… (More)