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 (Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin. 33 (2012) 1958--1964) is confirmed.Comment: Title slightly changed. 13 pages, to appear in European J. Combi

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Let G be an edge-colored graph. We use e(G) and c(G) to denote the number of edges of G and the number of colors appearing on E(G), respectively. For a vertex v ∈ V (G), the color neighborhood of v is defined as the set of colors assigned to the edges incident to v. A subgraph of G is rainbow if all of its edges are assigned with distinct colors. The well-known Mantel's theorem states that a graph G on n vertices contains a triangle if e(G) ≥ ⌊ n 2 4 ⌋ + 1. Rademacher (1941) showed that G contains at least ⌊ n 2 ⌋ triangles under the same condition. Li, Ning, Xu and Zhang (2014) proved a rainbow version of Mantel's theorem: An edge-colored graph G has a rainbow triangle if e(G) + c(G) ≥ n(n + 1)/2. In this paper, we first characterize all graphs G satisfying e(G) + c(G) ≥ n(n + 1)/2 − 1 but containing no rainbow triangles.Motivated by Rademacher's theorem, we then characterize all graphs G which satisfy e(G) + c(G) ≥ n(n + 1)/2 but contain only one rainbow triangle. We further obtain two results on color neighborhood conditions for the existence of rainbow short cycles.Our results improve a previous theorem due to Broersma, Li, Woeginger, and Zhang (2005). Moreover, we provide a sufficient condition in terms of color neighborhood for the existence of a specified number of vertex-disjoint rainbow cycles.

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