A Ramsey Property of Random Regular and $k$-out Graphs

TL;DR

This paper investigates Ramsey properties of random regular and $k$-out graphs, showing that even with edge colorings, certain monochromatic structures are either small or large, depending on the degree and coloring scheme.

Contribution

It establishes new Ramsey-type results for random regular and $k$-out graphs, revealing how their structure influences monochromatic component sizes and cycle lengths.

Findings

In even-degree regular graphs, edges can be colored so all monochromatic components are small.

In odd-degree regular graphs, any coloring yields a large monochromatic cycle.

Analogous results are proved for random $k$-out graphs.

Abstract

In this note we consider a Ramsey property of random $d$-regular graphs, $\mathcal{G}(n,d)$. Let $r\ge 2$ be fixed. Then w.h.p. the edges of $\mathcal{G}(n, 2r)$ can be colored such that every monochromatic component has size $o(n)$. On the other hand, there exists a constant $\gamma > 0$ such that w.h.p., every $r$-coloring of the edges of $\mathcal{G}(n, 2r+1)$ must contain a monochromatic cycle of length at least $\gamma n$. We prove an analogous result for random $k$-out graphs.