Vertex-Disjoint Rainbow Triangles In Edge-Colored Graphs

Let G be an edge-colored graph of order n. The minimum color degree of G, denoted by delta(c)(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 delta(c)(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 delta(c)(G) >= (n + 2)/2, then G contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if delta(c)(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 delta(c)(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. (c) 2020 Elsevier B.V. All rights reserved.