Covering spaces of locally homogeneous graphs

Abstract A graph G is called locally homogeneous, or locally G 0 , if for each vertex u of G the subgraph induced on the set of vertices adjacent with u is isomorphic to some graph G 0 . In this paper we use the concept of covering spaces for deriving various results on the set of all connected locally G 0 graphs (for given G 0 ). For instance, we prove that if e ( G 0 ) is small and there exists a locally G 0 graph, then there are infinitely many finite connected locally G 0 graphs. Further, a sufficient condition for existence of an infinite locally G 0 graph given in terms of minors of G is presented. As a by-product, we obtain a characterization of contraction minimal locally cyclic triangulations of the projective plane, which is also interesting for its own sake.