Rainbow cycles in edge-colored graphs

Let GG be a graph of order nn with an edge coloring cc, and let δc(G)δc(G) denote the minimum color degree of GG, i.e., the largest integer such that each vertex of GG is incident with at least δc(G)δc(G) edges having pairwise distinct colors. A subgraph F⊂GF⊂G is rainbow if all edges of FF have pairwise distinct colors. In this paper, we prove that (i) if GG is triangle-free and δc(G)>n3+1, then GG contains a rainbow C4C4, and (ii) if δc(G)>n2+2, then GG contains a rainbow cycle of length at least 4.