Rainbow cycles vs. rainbow paths

TL;DR

This paper explores the maximum number of rainbow subgraphs in edge-colored graphs avoiding certain rainbow cycles and paths, providing new bounds and exact results for specific cases, advancing understanding in rainbow Turán problems.

Contribution

It introduces bounds and exact results for rainbow copies of cycles and paths, notably improving bounds on rainbow paths and analyzing the case for cycles and paths of length 3, 4, and 5.

Findings

Established a new upper bound on $ ext{ex}^*(n,P_5)$.

Provided exact results for $ ext{ex}^*(n,C_ ext{ell})$ and $ ext{ex}^*(n,P_ ext{ell})$ for $ ext{ell} = 3,4,5$.

Enhanced bounds on rainbow paths, especially $ ext{ex}^*(n,P_ ext{ell})$.

Abstract

An edge-colored graph $F$ is {\it rainbow} if each edge of $F$ has a unique color. The {\it rainbow Tur\'an number} $\mathrm{ex}^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with no rainbow copy of $F$. The study of rainbow Tur\'an numbers was introduced by Keevash, Mubayi, Sudakov, and Verstra\"ete. Johnson and Rombach introduced the following rainbow-version of generalized Tur\'an problems: for fixed graphs $H$ and $F$, let $\mathrm{ex}^*(n,H,F)$ denote the maximum number of rainbow copies of $H$ in an $n$-vertex properly edge-colored graph with no rainbow copy of $F$. In this paper we investigate the case $\mathrm{ex}^*(n,C_\ell,P_\ell)$ and give a general upper bound as well as exact results for $\ell = 3,4,5$. Along the way we establish a new best upper bound on $\mathrm{ex}^*(n,P_5)$. Our main motivation comes from an attempt to improve bounds on $\mathrm{ex}^*(n,P_\ell)$, which has been the subject of several recent manuscripts.