Recent developments in half-transitive graphs

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 speciied, nite and undirected. For group-theoretic terms not deened here we refer the reader to 25]. If X is a graph we let V (X), E(X) and Aut X denote the sets of vertices and edges and the automorphism group of X, respectively. A subgroup G Aut X is said to be vertex-transitive, edge-transitive and arc-transitive provided it acts transitively on the sets of vertices, edges and arcs of X, respectively. Moreover, the subgroup G is said to be 1 2-transitive if it is vertex and edge but not arc-transitive. The graph X is said to be vertex-transitive, edge-transitive and arc-transitive if Aut X is vertex-transitive, edge-transitive and arc-transitive, respectively. We shall say that X is (G; 1 2)-transitive if G is a 1 2-transitive subgroup of automorphisms. In particular X is 1 2-transitive 1 Supported in part by \Ministrstvo za znanost in tehnologijo Slovenije", proj.no. J1-7035-94.