Rainbow spanning structures in strongly edge-colored graphs
Laihao Ding, Xiaolan Hu, Suyun Jiang
TL;DR
We address the problem of finding rainbow Hamilton cycles in strongly edge-colored graphs under minimum degree constraints. The key technique is to connect strongly edge-colored graphs to μn-bounded graphs and apply a switching-based lifting method to obtain rainbow Hamilton cycles in the bounded setting. The main results show that for all sufficiently large $n$, every strongly edge-colored $n$-vertex graph with $ obreak ext{delta}(G) obreak ight) rac{n+1}{2}$ contains a rainbow Hamilton cycle, and we characterize the extremal case at $ obreak ext{delta}(G)= rac{n}{2}$; as applications, we obtain optimal minimum-degree conditions for rainbow Hamilton paths and rainbow perfect matchings, confirming three conjectures for large $n$.
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 $μn$-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 $n$.