An Edge-Colored Version of Dirac's Theorem

Abstract

Let $G$ be an edge-colored graph. The minimum color degree $\delta^c(G)$ of $G$ is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on edges incident to $v$. We say that $G$ is properly colored if no two adjacent edges have the same color. In this paper, we show that every edge-colored graph $G$ with $\delta^c(G) \ge 2|G|/3$ contains a properly colored $2$-factor. Furthermore, we show that for any $\varepsilon > 0 $ there exists an integer $n_0$ such that every edge-colored graph $G$ with $|G| = n \ge n_0$ and $\delta^c(G) \ge ( 2/3 + \varepsilon ) n $ contains a properly colored cycle of length $\ell$ for every $3 \le \ell \le n$. This result is best possible in the sense that the statement is false for $\delta^c(G) < 2n/3$.