Note on rainbow cycles in edge-colored graphs

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Let G be a graph of order n with an edge-coloring c , and let δ c ( G ) denote the minimum color-degree of G . A subgraph F of G is called rainbow if all edges of F have pairwise distinct colors. There have been a lot of results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if δ c ( G ) > 2 n − 1 3 , then every vertex of G is contained in a rainbow triangle; (ii) if δ c ( G ) > 2 n − 1 3 and n ≥ 13 , then every vertex of G is contained in a rainbow C 4 ; (iii) if G is complete, n ≥ 7 k − 17 and δ c ( G ) > n − 1 2 + k , then G contains a rainbow cycle of length at least k , where k ≥ 5 .