Matroidal bijections between graphs

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_n², 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.