Partitioning random graphs into monochromatic components

TL;DR

This paper investigates how random graphs can be partitioned into monochromatic components under various coloring schemes, extending classical conjectures and results from complete graphs to sparse random graphs.

Contribution

It extends known results on monochromatic partitions from complete graphs to random graphs with large minimum degree and sparse regimes, providing new thresholds and partial analogs.

Findings

For p ≥ (27 log n / n)^{1/3}, a.a.s. every 2-coloring of G(n,p) has a partition into two monochromatic components.

For r ≥ 2 and p ≪ (r log n / n)^{1/r}, a.a.s. there exists an r-coloring with no bounded component cover.

If p = ω(1)/n, then a.a.s. every r-coloring of G(n,p) has a large monochromatic component of size at least (1-o(1)) n/(r-1).

Abstract

Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every $r$-colored complete graph can be partitioned into at most $r-1$ monochromatic components; this is a strengthening of a conjecture of Lov\'asz (1975) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into $r$ monochromatic components is possible for sufficiently large $r$-colored complete graphs. We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if $p\ge \left(\frac{27\log n}{n}\right)^{1/3}$, then a.a.s. in every $2$-coloring of $G(n,p)$ there exists a partition into two monochromatic components, and for $r\geq 2$ if $p\ll \left(\frac{r\log n}{n}\right)^{1/r}$, then a.a.s. there exists an $r$-coloring of $G(n,p)$ such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gy\'arf\'as (1977) about large monochromatic components in $r$-colored complete graphs. We show that if $p=\frac{\omega(1)}{n}$, then a.a.s. in every $r$-coloring of $G(n,p)$ there exists a monochromatic component of order at least $(1-o(1))\frac{n}{r-1}$.