Rainbow triangles in edge-colored graphs

TL;DR

This paper establishes conditions based on color degree, color number, and edge number that guarantee the existence of rainbow triangles in edge-colored graphs, confirming a conjecture by Li and Wang.

Contribution

It provides new sufficient conditions for rainbow triangle existence and confirms a previously proposed conjecture in the field.

Findings

Sufficient conditions for rainbow triangles based on color degree and number

Confirmation of Li and Wang's conjecture

Enhanced understanding of rainbow triangle existence criteria

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.