Rainbow spanning structures in strongly edge-colored graphs

TL;DR

This paper proves that large strongly edge-colored graphs with high minimum degree contain rainbow Hamilton cycles, characterizes exceptions, and establishes optimal conditions for rainbow paths and matchings, confirming three conjectures.

Contribution

It establishes minimum degree conditions for rainbow Hamilton cycles, paths, and matchings in strongly edge-colored graphs, confirming three conjectures for large graphs.

Findings

High minimum degree ensures rainbow Hamilton cycles

Characterization of graphs without rainbow Hamilton cycles at degree n/2

Optimal degree conditions for rainbow paths and matchings

Abstract

An edge-colored graph is a graph in which each edge is assigned a color. Such a graph is called strongly edge-colored if each color class forms an induced matching, and called rainbow if all edges receive pairwise distinct colors. In this paper, by establishing a connection with $\mu n$-bounded graphs, we prove that for all sufficiently large integers $n$, every strongly edge-colored graph $G$ on $n$ vertices with minimum degree at least $\frac{n+1}{2}$ contains a rainbow Hamilton cycle. We also characterize all strongly edge-colored graphs on $n$ vertices with minimum degree exactly $\frac{n}{2}$ that do not contain a rainbow Hamilton cycle. As an application, we determine the optimal minimum degree conditions for the existence of rainbow Hamilton paths and rainbow perfect matchings in strongly edge-colored graphs. Together, these results verify three conjectures concerning strongly edge-colored graphs for sufficiently large $n$.