Let p be a prime. It was shown by Folkman (J. Combin. Theory 3 (1967) 215) that a regular edge-transitive graph of order 2p or 2p is necessarily vertex-transitive. In this paper an extension of his result in the case of cubic graphs is given. It is proved that, with the exception of the Gray graph on 54 vertices, every cubic edge-transitive graph of order 2p is vertex-transitive. c © 2003 Elsev...

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.

Finite graphs of valency 4 and girth 4 admitting 1 2-transitive group actions, that is vertex-and edge-but not arc-transitive group actions, are investigated. A graph is said to be 1 2-transitive if its automorphism group acts 1 2-transitively. There is a natural (balanced) orientation of the edge set of a 1 2-transitive graph induced and preserved by its auto-morphism group. Among others it is...

We develop a new framework for analysing finite connected, oriented graphs of valency four, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of ‘basic’ graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms t...

In this paper, we prove that every vertex-transitive graph can be expressed as the edge-disjoint union of symmetric graphs. We define a multicycle graph and conjecture that every vertex-transitive graph cam be expressed as the edge-disjoint union of multicycles. We verify this conjecture for several subclasses of vertextransitive graphs, including Cayley graphs, multidimensional circulants, and...