An edge-colored version of Dirac's theorem

Let G be an edge-colored graph. The minimum color degree δ^c(G) of G is the largest integer k such that for every vertex ν, there are at least k distinct colors on edges incident to ν. 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 δ^c(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 δ^c(G) ≥ (2/3 + ε)n contains a properly colored cycle of length ℓ for every 3 ≤ ℓ ≤ n. This result is best possible in the sense that the statement is false for δ^c(G) < 2n/3.