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

If G is a graph and Eta is a set of subgraphs of G, we say that an edge-coloring of G is Eta-polychromatic if every graph from Eta gets all colors present in G on its edges. The Eta-polychromatic number of G, denoted by polyH(G), is the largest number of colors in an Eta-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 Eta 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. (c) 2022 Elsevier B.V. All rights reserved.