Let G be a group acting symmetrically on a graph 2, let G, be a subgroup of G minimal among those that act symmetrically on 8, and let G2 be a subgroup of G, maximal among those normal subgroups of GI which contain no member except 1 which fixes a vertex of Z. The most precise result of this paper is that if Z has prime valency p , then either Z is a bipartite graph or G2 acts regularly on Z or GI I G2 is a simple group which acts symmetrically on a graph of valency p which can be constructed from C and does not have more vertices than 2. The results on vertextransitive groups necessary to establish results like this are also included.
If G1,…,Gc are graphs without loops or multiple edges there is a smallest integer r(G1,…,Gc) such that if the edges of a complete graph Kn, with n ≧ r(G1,…,Gc), are painted arbitrarily with c colours the ith coloured subgraph contains Gi as a subgraph for at least one i. r(G1,…Gc) is called the Ramsey number of the graphs G1,…,Gc.
A number of constructions are given for arc-transitive digraphs, based on modifications of permutation representations of finite groups. In particular, it is shown that for every positive integer .s and for any transitive permutation group P of degree k, there are infinitely many examples of a finite k-regular digraph with a group of automorphisms acting transitively on s-arcs (but not on (s + l)-arcs), such that the stabilizer of a vertex induces the action of P on the out-neighbour set.1991 Mathematics subject classification (Amer. Math. Soc): 05C25, 20B25.
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