Vertex-disjoint rainbow triangles in edge-colored graphs

Abstract

Let G be an edge-colored graph of order n. The minimum color degree of G, denoted by δc(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 an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if δc(G)≥(n+1)∕2, then G contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if n≥20 and δc(G)≥(n+2)∕2, then G contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if δc(G)≥(n+k)∕2, then G contains k vertex-disjoint rainbow triangles. For any integer k≥2, we show that if n≥16k−12 and δc(G)≥n∕2+k−1, then G contains k vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of k edge-disjoint rainbow triangles.