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 by polyH(G), is the largest number of colors in an H-polychromatic coloring. In this paper we determine polyH(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.
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