Polychromatic colorings of complete graphs with respect to 1-, 2-factors and Hamiltonian cycles.

If G is a graph and H is a set of subgraphs of G, then an edge-coloring of G is called H-polychromatic if every graph from H gets all colors present in G. The H-polychromatic number of G, denoted poly H(G), is the largest number of colors such that G has an H-polychromatic coloring. In this article, poly H(G) is determined exactly when G is a complete graph and H is the family of all 1-factors. In addition polyH(G) is found up to an additive constant term when G is a complete graph and H is the family of all 2-factors, or the family of all Hamiltonian cycles.