Rainbow Cliques in Edge-Colored Graphs

TL;DR

This paper extends a known result about rainbow triangles in edge-colored graphs to larger cliques, establishing conditions under which rainbow $K_r$ subgraphs must exist.

Contribution

The paper generalizes Li's theorem from triangles to larger cliques, providing new minimum color degree conditions for rainbow $K_r$ existence.

Findings

For $r \\ge 4$, minimum color degree conditions guarantee rainbow $K_r$.

Extended Li's result from triangles to larger cliques.

Provides new bounds for rainbow clique existence in edge-colored graphs.

Abstract

Let $G = (V,E)$ be an $n$-vertex graph and let $c: E \to \mathbb{N}$ be a coloring of its edges. Let $d^c(v)$ be the number of distinct colors on the edges at $v \in V$ and let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$. H. Li proved that $\delta^c(G) > n/2$ guarantees a rainbow triangle in $G$. We give extensions of Li's result to cliques $K_r$ for $r \ge 4$.