Properly colored short cycles in edge-colored graphs

TL;DR

This paper explores the existence of properly colored cycles in edge-colored graphs, establishing conditions under which such cycles of bounded length exist, and relating these to the Caccetta-H"{a}ggkvist Conjecture.

Contribution

It proves a new relation between properly colored cycles and directed cycles, and links the existence of such cycles to the Caccetta-H"{a}ggkvist Conjecture under certain degree conditions.

Findings

Existence of spanning subgraphs with properly colored cycles under no $K_{s,t}$ condition

Conditional bounds on cycle length based on the Caccetta-H"{a}ggkvist Conjecture

Asymptotically tight degree conditions for properly colored $K_{s,t}$

Abstract

Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-H\"{a}ggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-colored graph. We first prove that for all integers $s$ and $t$ with $t\geq s\geq2$, every edge-colored graph $G$ with no properly colored $K_{s,t}$ contains a spanning subgraph $H$ which admits an orientation $D$ such that every directed cycle in $D$ is a properly colored cycle in $G$. Using this result, we show that for $r\geq4$, if the Caccetta-H\"{a}ggkvist Conjecture holds , then every edge-colored graph of order $n$ with minimum color degree at least $n/r+2\sqrt{n}+1$ contains a properly colored cycle of length at most $r$. In addition, we also obtain an asymptotically tight total color degree condition which ensures a properly colored (or rainbow) $K_{s,t}$.