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arXiv · 2601.16084

Rainbow spanning structures in strongly edge-colored graphs

Laihao Ding·Xiaolan Hu·Suyun Jiang
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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 $n$.

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Laihao Ding, Xiaolan Hu, Suyun Jiang. 2026-01-22. Rainbow spanning structures in strongly edge-colored graphs. https://arxiv.org/abs/2601.16084

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