An Edge-Colored Version of Dirac's Theorem

Let G be an edge-colored graph. The minimum color degree δᶜ(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 δᶜ(G) ≥ 2|G|/3 contains a properly colored $2$-factor. Furthermore, we show that for any ε > 0 there exists an integer n₀ such that every edge-colored graph G with |G| = n ≥ n₀ and δᶜ(G) ≥ ( 2/3 + ε ) n contains a properly colored cycle of length for every 3 ≤ ≤ n. This result is best possible in the sense that the statement is false for δᶜ(G) < 2n/3.