Note on rainbow cycles in edge-colored graphs

Abstract

Let G be a graph of order n with an edge-coloring c, and let δc(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 δc(G)>2n−13, then every vertex of G is contained in a rainbow triangle; (ii) if δc(G)>2n−13 and n≥13, then every vertex of G is contained in a rainbow C4; (iii) if G is complete, n≥7k−17 and δc(G)>n−12+k, then G contains a rainbow cycle of length at least k, where k≥5.