Long monochromatic paths and cycles in 2-colored bipartite graphs

TL;DR

This paper extends classical results on monochromatic paths and cycles in bipartite graphs to non-complete graphs with high minimum degree, establishing stability and tight bounds for monochromatic cycle lengths.

Contribution

It proves a stability version of the monochromatic path and cycle existence in nearly complete bipartite graphs and determines tight bounds on cycle lengths based on minimum degree.

Findings

Monochromatic cycle of length at least (1+o(1))n exists under certain conditions.

Characterization of extremal colorings close to the bounds.

Asymptotically tight bounds for monochromatic cycle lengths based on minimum degree.

Abstract

Gy\'arf\'as and Lehel and independently Faudree and Schelp proved that in any 2-coloring of the edges of $K_{n,n}$ there exists a monochromatic path on at least $2\lceil n/2\rceil$ vertices, and this is tight. We prove a stability version of this result which holds even if the host graph is not complete; that is, if $G$ is a balanced bipartite graph on $2n$ vertices with minimum degree at least $(3/4+o(1))n$, then in every 2-coloring of the edges of $G$, either there exists a monochromatic cycle on at least $(1+o(1))n$ vertices, or the coloring of $G$ is close to an extremal coloring -- in which case $G$ has a monochromatic path on at least $2\lceil n/2\rceil$ vertices and a monochromatic cycle on at least $2\lfloor n/2\rfloor$ vertices. Furthermore, we determine an asymptotically tight bound on the length of a longest monochromatic cycle in a 2-colored balanced bipartite graph on $2n$ vertices with minimum degree $\delta n$ for all $0\leq \delta\leq 1$.