Covering Regular Graphs

A covering projection from a graph G onto a graph H is a \local isomorphism": a mapping from the vertex set of G onto the vertex set of H such that, for every v 2 V (G), the neighborhood of v is mapped bijectively onto the neighborhood (in H) of the image of v. We investigate two concepts that concern graph covers of regular graphs. The rst one is called \multicovers": we show that for any regular graph H there exists a graph G that allows many diierent covering projections onto H. Secondly, we consider partial covers, which require only that G be a subgraph of a cover of H. As an application of our results we show that there are innnitely many rigid regular graphs H for which the H-cover problem { deciding if a given graph G covers H { is NP-complete. This resolves an open problem related to the characterization of graphs H for which H-COVER is tractable. 1 Motivation and overview For a xed graph H, the H-cover problem admits a graph G as input and asks about the existence of a \local isomorphism": a labeling of vertices of G by vertices of H so that the label set of the neighborhood of every v 2 V (G) is equal to the neighborhood (in H) of the label of v and each neighbor of v is labeled by a diierent neighbor of the label of v. (Such a labeling is referred to as a covering projection from G onto H.) We trace this concept to Conway and Biggs's construction of innnite classes of highly symmetric graphs, see Chapter 19 of 4]. Graph coverings are special cases of covering spaces from algebraic topology 17], and are used in many applications in topological graph theory