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 (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

Properly colored cycles in edge-colored graphs

“…In 2013, H. Li [39] confirmed this conjecture himself. Independently, B. Li et al [38] verified this conjecture by proving a stronger result, and they also characterized the corresponding extremal graphs.…”

“…Li et al [38] also proved that the bound of color degree in Theorem 2.1 is tight for the existence of rainbow triangles, but can be lowered to n/2 with some simple exceptions.…”

“…In [38], Li et al also gave an "|E(G)| + |col(G)|" condition for the existence of rainbow triangles in an edge-colored graph G. , then G contains a rainbow triangle.…”

“…We recommend the proof techniques in [38] and [46], and the constructions in [28] and Chapter 16 of [12] for further details on the close relationship between edge-colored graphs and directed graphs. The last three chapters in this thesis are motivated by and dedicated to this relationship.…”

“…For short PC cycles, especially, a PC triangle (or a rainbow triangle), the well-known Gallai coloring theory gives a structural characterization of edge-colored complete graphs containing no rainbow triangles (See [32] and [34]). Conditions for the existence of rainbow triangles in edge-colored graphs (not necessarily complete) are given in [38] and [39]. In a variety of those work, minimum color degree conditions for edge-colored graphs to contain PC cycles with certain properties are often discussed.…”