A new family of expansive graphs

Abstract

An affine graph is a pair (G, ) where G is a graph and is an automorphism assigning to each vertex of G one of its neighbors. On one hand, we obtain a structural decomposition of any affine graph (G, ) in terms of the orbits of . On the other hand, we establish a relation between certain colorings of a graph G and the intersection graph of its cliques K(G). By using the results we construct new examples of expansive graphs. The expansive graphs were introduced by Neumann-Lara in 1981 as a stronger notion of the K-divergent graphs. A graph G is K-divergent if the sequence |V (Kn(G))| tends to infinity with n, where Kn+1(G) is defined by Kn+1(G)=K(Kn(G)) for n 1. In particular, our constructions show that for any k 2, the complement of the Cartesian product Ck, where C is the cycle of length 2k +1, is K-divergent.