Rainbow triangles in edge-colored complete graphs
Xiaozheng Chen, Xueliang Li

TL;DR
This paper generalizes and sharpens existing results on the existence and quantity of rainbow triangles in edge-colored complete graphs, establishing new minimum color-degree conditions for such structures.
Contribution
It extends previous theorems by providing broader minimum color-degree bounds that guarantee rainbow triangles and multiple disjoint rainbow triangles in complete graphs.
Findings
Every vertex in a complete graph with minimum color-degree at least (n+k)/2 is in at least k rainbow triangles.
Complete graphs with minimum color-degree at least n/2 contain a rainbow triangle, and this bound is optimal.
For n ≥ 7, such graphs contain two vertex-disjoint rainbow triangles, improving previous bounds.
Abstract
Let be a graph of order with an edge-coloring , and let denote the minimum color-degree of . A subgraph of is called rainbow if any two edges of have distinct colors. There have been a lot results in the existing literature on rainbow triangles in edge-colored complete graphs. Fujita and Magnant showed that for an edge-colored complete graph of order , if , then every vertex of is contained in a rainbow triangle. In this paper, we show that if , then every vertex of is contained in at least rainbow triangles, which can be seen as a generalization of their result. Li showed that for an edge-colored graph of order , if , then contains a rainbow triangle. We show that if is complete and , then …
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
