Long monochromatic paths and cycles in 2-edge-colored multipartite graphs

TL;DR

This paper characterizes conditions for the existence of large monochromatic paths and cycles in 2-edge-colored complete multipartite graphs, extending known conjectures and utilizing a stability theorem on monochromatic connected matchings.

Contribution

It provides a complete description of when large monochromatic paths and cycles must exist in 2-edge-colored complete multipartite graphs, generalizing previous conjectures.

Findings

Characterization of all $n_1,...,n_s$ for large $n$ ensuring monochromatic paths and cycles.

Extension of Gyárfás-Ruszinkó-Sárközy-Szemerédi conjecture to multipartite graphs.

Application of a new stability theorem on monochromatic connected matchings.

Abstract

We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i) cycle $C_{2n}$ with $2n$ vertices, (ii) cycle $C_{\geq 2n}$ with at least $2n$ vertices, (iii) path $P_{2n}$ with $2n$ vertices, and (iv) path $P_{2n+1}$ with $2n+1$ vertices. This implies a generalization for large $n$ of the conjecture by Gy\'arf\'as, Ruszink\'o, S\'ark\H{o}zy and Szemer\'edi that for every $2$-edge-coloring of the complete $3$-partite graph $K_{n,n,n}$ there is a monochromatic path $P_{2n+1}$. An important tool is our recent stability theorem on monochromatic connected matchings.