A transitive orientation of an undirected graph is an assignment of directions to itsedges so that these directed edges represent a transitive relation between the vertices ofthe graph. Not every graph has a transitive orientation, but every graph can be turnedinto a graph that has a transitive orientation, by adding edges. We study the problem ofadding an inclusion minimal set ...

In this note connected, edge-transitive lexicographic and Cartesian products are characterized. For the lexicographic product G ◦H of a connected graph G that is not complete by a graph H, we show that it is edge-transitive if and only if G is edge-transitive and H is edgeless. If the first factor of G ◦H is non-trivial and complete, then G ◦H is edge-transitive if and only if H is the lexicogr...

A construction is given of a valent arc transitive graph with vertex stabilizer isomorphic to the dihedral group D The graph has vertices and is the rst known example of a valent arc transitive graph with nonabelian vertex stabilizer A VALENT HALF ARC TRANSITIVE GRAPH

A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. Let p be a prime. Cheng and Oxley [On weakly symmetric graphs of order twice a prime, J. Combin. Theory B 42(1987) 196-211] proved that there is no half-arc-transitive graph of order 2p, and Alspach and Xu [ 12 -transitive graphs of order 3p, J. Algebraic Combin. 3(1994) 347-355] c...

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...

Every finite, self-dual, regular (or chiral) 4-polytope of type {3, q, 3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edgebut not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.

Slovenija Abstract A vertex-transitive graph is said to be 1 2-transitive if its automor-phism group is vertex and edge but not arc-transitive. Some recent results on 1 2-transitive graphs are given, a few open problems proposed , and possible directions in the research of the structure of these graphs discussed. 1 In the beginning Throughout this paper graphs are simple and, unless otherwise s...

A cycle decomposition of a graph Γ is a set C of cycles of Γ such that every edge of Γ belongs to exactly one cycle in C. Such a decomposition is called arc-transitive if the group of automorphisms of Γ that preserve C setwise acts transitively on the arcs of Γ . In this paper, we study arc-transitive cycle decompositions of tetravalent graphs. In particular, we are interested in determining an...

A subgroup of the automorphism group a graph acts half-arc-transitively on if it transitively vertex-set and edge-set but not arc-set graph. If full half-arc-transitively, is said to be half-arc-transitive. In 1994 Gardiner Praeger introduced two families tetravalent arc-transitive graphs, called C±1 C±ε that play prominent role in characterization graphs admitting an automorphisms with normal ...

Let φ be Euler’s phi function. We prove that a vertex-transitive graph 0 of order n, with gcd(n, φ(n)) = 1, is isomorphic to a circulant graph of order n if and only if Aut(0) contains a transitive solvable subgroup. As a corollary, we prove that every vertex-transitive graph 0 of order n is isomorphic to a circulant graph of order n if and only if for every such 0, Aut(0) contains a transitive...