Polychromatic colorings of 1-regular and 2-regular subgraphs of complete graphs

Abstract

If G is a graph and H is a set of subgraphs of G, we say that an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted poly H (G), is the largest number of colors in an Hpolychromatic coloring. In this paper we determine poly H (G) exactly when G is a complete graph on n vertices, q is a fixed nonnegative integer, and H is one of three families: the family of all matchings spanning n − q vertices, the family of all 2-regular graphs spanning at least n − q vertices, and the family of all cycles of length precisely n − q. There are connections with an extension of results on Ramsey numbers for cycles in a graph.