C O ] 2 7 D ec 2 01 2 Heterochromatic triangles in edge-colored graphs ∗

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 heterochromatic triangle is one in which every pair of edges have different colors. In this paper we give some sufficient conditions for the existence of heterochromatic 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 (Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin. 33 (2012) 1958-1964) is confirmed.