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

TL;DR

This paper determines the maximum number of colors in edge-colorings of complete graphs such that specific subgraph families (matchings, 2-regular graphs, cycles) contain all colors, extending results related to cycle Ramsey numbers.

Contribution

It provides exact values of the $ ext{poly}_ ext{H}(G)$ for complete graphs with respect to three specific subgraph families, linking to cycle Ramsey number extensions.

Findings

Exact $ ext{poly}_ ext{H}(G)$ values for matchings, 2-regular graphs, and cycles.

Connections established between polychromatic colorings and cycle Ramsey numbers.

Results apply to complete graphs with fixed parameters and subgraph sizes.

Abstract

If $G$ is a graph and $\mathcal{H}$ is a set of subgraphs of $G$, we say that an edge-coloring of $G$ is $\mathcal{H}$-polychromatic if every graph from $\mathcal{H}$ gets all colors present in $G$ on its edges. The $\mathcal{H}$-polychromatic number of $G$, denoted $\operatorname{poly}_\mathcal{H} (G)$, is the largest number of colors in an $\mathcal{H}$-polychromatic coloring. In this paper we determine $\operatorname{poly}_\mathcal{H} (G)$ exactly when $G$ is a complete graph on $n$ vertices, $q$ is a fixed nonnegative integer, and $\mathcal{H}$ 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.