Rainbow Tur\'an Problem for Even Cycles

TL;DR

This paper investigates the maximum number of edges in properly edge-colored graphs that avoid rainbow even cycles, establishing bounds that relate graph density to the guaranteed existence of such cycles.

Contribution

It provides new bounds on the rainbow Turán number for even cycles, partially answering a question about the density threshold for rainbow cycle existence.

Findings

Graphs with sufficiently many edges contain rainbow even cycles of bounded length.

The paper establishes a relationship between graph density and the guaranteed presence of rainbow cycles.

It introduces bounds involving logarithmic functions of the graph's density parameter.

Abstract

An edge-colored graph is rainbow if all its edges are colored with distinct colors. For a fixed graph $H$, the rainbow Tur\'an number $\mathrm{ex}^{\ast}(n,H)$ is defined as the maximum number of edges in a properly edge-colored graph on $n$ vertices with no rainbow copy of $H$. We study the rainbow Tur\'an number of even cycles, and prove that for every fixed $\varepsilon > 0$, there is a constant $C(\varepsilon)$ such that every properly edge-colored graph on $n$ vertices with at least $C(\varepsilon) n^{1 + \varepsilon}$ edges contains a rainbow cycle of even length at most $2 \lceil \frac{\ln 4 - \ln \varepsilon}{\ln (1 + \varepsilon)} \rceil$. This partially answers a question of Keevash, Mubayi, Sudakov, and Verstra\"ete, who asked how dense a graph can be without having a rainbow cycle of any length.