Matroidal bijections between graphs

Abstract

We study a hierarchy of five classes of bijections between the edge sets of two graphs: weak maps, strong maps, cyclic maps, orientable cyclic maps, and chromatic maps. Each of these classes contains the next one and is a natural class of mappings for some family of matroids. For example, f: E( G) → E( H) is cyclic if every cycle (eulerian subgraph) of G is mapped onto a cycle of H. This class of mappings is natural when graphs are considered as binary matroids. A chromatic map E( G) → E( H) is induced by a (vertex) homomorphism from G to H. For such maps, the notion of a vertex is meaningful so they are natural for graphic matroids. In the same way that chromatic maps lead to the definition of χ( G)—the chromatic number—the other classes give rise to new interesting graph parameters. For example, φ( G) is the least order of H for which there exists a cyclic bijection f: E( G) → E( H). We establish some connection between φ and χ, e.g., χ(G) ≥ φ(G) > χ(G) 2 . The exact relation between φ and χ depends on knowledge of the chromatic number of C n 2, the square of the n-dimensional cube. Higher powers of C n are considered, too, and tight bounds for their chromatic number are found, through some analysis of their eigenvalues.