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

TL;DR

This paper determines the maximum number of colors in edge-colorings of complete graphs that ensure all 1-factors, 2-factors, or Hamiltonian cycles contain all colors, advancing understanding of graph colorings in combinatorics.

Contribution

It provides exact values for the H-polychromatic number when G is complete and H is all 1-factors, and bounds for 2-factors and Hamiltonian cycles.

Findings

Exact poly_H(G) for 1-factors in complete graphs.

Bounds for poly_H(G) for 2-factors and Hamiltonian cycles.

Advances in understanding colorings ensuring subgraph color diversity.

Abstract

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 on its edges. The H-polychromatic number of G, denoted poly_H(G), is the largest number of colors in an H-polychromatic coloring. In this paper, poly_H(G) is determined exactly when G is a complete graph and H is the family of all 1-factors. In addition poly_H(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.