Rainbow triangles in edge-colored graphs

Abstract

Let G be an edge-colored graph. The color degree of a vertex v of G, is defined as the number of colors of the edges incident to v. The color number of G is defined as the number of colors of the edges in G. A rainbow triangle is one in which every pair of edges have distinct colors. In this paper we give some sufficient conditions for the existence of rainbow triangles in edge-colored graphs in terms of color degree, color number and edge number. As a corollary, a conjecture proposed by Li and Wang [H. Li and G. Wang, Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin. 33 (2012) 1958–1964] is confirmed.