M ar 2 00 7 A 27 - vertex graph that is vertex - transitive and edge - transitive but not l - transitive

I describe a 27-vertex graph that is vertex-transitive and edgetransitive but not 1-transitive. Thus while all vertices and edges of this graph are similar, there are no edge-reversing automorphisms. A graph (undirected, without loops or multiple edges) is said to be vertextransitive if its automorphism group acts transitively on the set of vertices, edge-transitive if its automorphism group acts transitively on the set of undirected edges, and 1-transitive if its automorphism group acts transitively on the set of paths of length 1. If a graph is edge-transitive but not 1-transitive then any edge can be mapped to any other, but in only one of the two possible ways. In my Harvard senior thesis [2], I described a graph that is vertextransitive and edge-transitive but not 1-transitive. It has 27 vertices, and is regular of degree 4. This beautiful graph was also discovered by Derek Holt [4]. It seems likely that this is the smallest graph that is vertex-transitive and edge-transitive but not 1-transitive. The question of the existence of graphs that are vertex-transitive and edge-transitive but not 1-transitive was raised by Tutte [5], who showed that any such graph must be regular of even degree. The first examples were given Derived from the Harvard senior thesis of Peter G. Doyle, dated June 1976. Copyright (C) 1976, 1985 Peter G. Doyle. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, as published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.