Note on rainbow cycles in edge-colored graphs

TL;DR

This paper investigates conditions on edge-colored graphs that guarantee the existence of rainbow cycles of various lengths, providing new bounds and correcting previous inaccuracies in the literature.

Contribution

It establishes new minimum color degree thresholds ensuring rainbow triangles, quadrilaterals, and long cycles, and corrects gaps in prior research.

Findings

Every vertex is in a rainbow triangle if elta^c(G)>rac{3n-3}{4}

Every vertex is in a rainbow C4 if elta^c(G)>rac{3n}{4}

Existence of long rainbow cycles in complete graphs with certain degree conditions

Abstract

Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $\delta^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if $\delta^c(G)>\frac{3n-3}{4}$, then every vertex of $G$ is contained in a rainbow triangle; (ii) $\delta^c(G)>\frac{3n}{4}$, then every vertex of $G$ is contained in a rainbow $C_4$; and (iii) if $G$ is complete, $n\geq 8k-18$ and $\delta^c(G)>\frac{n-1}{2}+k$, then $G$ contains a rainbow cycle of length at least $k$. Some gaps in previous publications are also found and corrected.