On Leighton’s Graph Covering Theorem

We give short expositions of both Leighton’s proof and the BassKulkarni proof of Leighton’s graph covering theorem, in the context of colored graphs. We discuss a further generalization, needed elsewhere, to “symmetryrestricted graphs.” We can prove it in some cases, for example, if the “graph of colors” is a tree, but we do not know if it is true in general. We show that Bass’s Conjugation Theorem, which is a tool in the Bass-Kulkarni approach, does hold in the symmetry-restricted context. Leighton’s graph covering theorem says: Theorem (Leighton [5]). Two finite graphs which have a common covering have a common finite covering. It answered a conjecture of Angluin and Gardiner who had proved the case that both graphs are k–regular [1]. Leighton’s proof is short (two pages), but has been considered by some to lack transparency. It was reframed in terms of Bass-Serre theory by Bass and Kulkarni [2, 3], expanding its length considerably but providing group-theoretic tools which have other uses. The general philosophy of the Bass-Kulkarni proof is that adding more structure helps. Let us illustrate this by giving a very short proof of Angluin and Gardiner’s original k–regular case. We assume all graphs considered are connected. “Graph” will thus mean a connected 1–complex. “Covering” means covering space in the topological sense. Two graphs are isomorphic if they are isomorphic as 1–complexes (i.e., homeomorphic by a map which is bijective on the vertex and edge sets). We want to show that if G and G′ are finite k-regular graphs (i.e., all vertices have valence k) then they have a common finite covering. Proof of the k–regular case. Replace G and G′ by oriented “fat graphs”—thicken edges to rectangles of length 10 and width 1, say, and replace vertices by regular planar k-gons of side length 1, to which the rectangles are glued at their ends (see Fig. 1; the underlying space of the fat graph is often required to be orientable as a 2-manifold but we don’t need this). G and G′ both have universal covering the k– regular fat tree Tk, whose isometry group Γ acts properly discretely (the orbit space Tk/Γ is the 2–orbifold pictured in Fig. 2). The covering transformation groups for the coverings Tk → G and Tk → G′ are finite index subgroups Λ and Λ′ of Γ. The quotient Tk/(Λ ∩ Λ′) is the desired common finite covering of G and G′. We return now to unfattened graphs. In addition to the simplicial view of graphs, it is helpful to consider in parallel a combinatorial point of view, in which an edge of an undirected graph consists of a pair (e, ē) of directed edges. From this point of view a graph G is defined by a vertex set V(G) and directed edge set E(G),