Regular graphs and stability

We show that a graph G is semi-stable at vertex v if and only if the set of vertices of G adjacent to v is fixed by the automorphism group of Gv, the subgraph of G obtained by deleting v and its incident edges. This result leads to a neat proof that regular graphs are semi-stable at each vertex. We then investigate stable regular graphs, concentrating mainly on stable vertex-transitive graphs. We conjecture that if G is a non-trivial vertex-transitive graph, then G is stable if and only if γ(G) contains a transposition, offering some evidence for its truth.

Cayley maps and symmetrical maps

In this paper we shall show how combinatorial methods can be applied to the study of maps on orientable surfaces. Our main concern is with maps which possess a certain kind of symmetry, called vertex-transitivity. We show how an extension of the well-known method of Cayley can be used to construct such maps, and we give conditions which suffice for the automorphism groups of these maps to have non trivial vertex-stabilizers. Finally, we investigate the special case when the skeleton of the map is a complete graph; a classical theorem of Frobenius then implies that all vertex-transitive maps are given by our extension of Cayley's construction.