Large monochromatic components in almost complete graphs and bipartite graphs

TL;DR

This paper improves bounds on the minimum degree condition needed to guarantee large monochromatic components in almost complete graphs under edge colorings, advancing understanding of monochromatic connectivity in dense graphs.

Contribution

It establishes a new, smaller bound for the minimum degree threshold, specifically /(6t^3), ensuring large monochromatic components in almost complete graphs, improving previous results.

Findings

/(6t^3) suffices for large monochromatic components

Improves previous bounds on minimum degree conditions

Advances understanding of monochromatic connectivity in dense graphs

Abstract

Gy\'arfas proved that every coloring of the edges of $K_n$ with $t+1$ colors contains a monochromatic connected component of size at least $n/t$. Later, Gy\'arf\'as and S\'ark\"ozy asked for which values of $\gamma=\gamma(t)$ does the following strengthening for almost complete graphs hold: if $G$ is an $n$-vertex graph with minimum degree at least $(1-\gamma)n$, then every $(t+1)$-edge coloring of $G$ contains a monochromatic component of size at least $n/t$. We show $\gamma = 1/(6t^3)$ suffices, improving a result of DeBiasio, Krueger, and S\'ark\"ozy.