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) --t 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 X(Gtthe chromatic number-the other classes give rise to new interesting graph parameters. For example, 4(G) is the least order of H for which there exists a cyclic bijection f: E(G) --t E(H). We establish some connection between 4 and x, e.g., x(G) > i(G) > x(G)/2. The exact relation between 4 and x depends on knowledge of the chromatic number of C$ the square of the n-dimensional cube. Higher powers of C, are considered, too, and tight bounds for their chromatic number are found, through some analysis of their eigenvalues.