The chromatic number of random lifts of complete graphs

TL;DR

This paper investigates the chromatic number of random lifts of complete graphs, establishing asymptotic concentration results and extending previous questions about the chromatic number in such graph constructions.

Contribution

It determines the asymptotic distribution of the chromatic number for random lifts of complete graphs and extends the analysis to general fixed regular graphs.

Findings

Chromatic number concentrates on either k or k+1 for fixed d≥3.

For about half of the values of d, the chromatic number is concentrated on k.

The upper bound proof uses the small subgraph conditioning method.

Abstract

An $n$-lift of a graph $G$ is a graph from which there is an $n$-to-$1$ covering map onto $G$. Amit, Linial, and Matou\v sek (2002) raised the question of whether the chromatic number of a random $n$-lift of $K_5$ is concentrated on a single value. We consider this problem for $G=K_{d+1}$, and show that for fixed $d\ge 3$ the chromatic number of a random lift of $K_d$ is (asymptotically almost surely) either $k$ or $k+1$, where $k$ is the smallest integer satisfying $d < 2k \log k$. Moreover, we show that, for roughly half of the values of $d$, the chromatic number is concentrated on $k$. The argument for the upper-bound on the chromatic number uses the small subgraph conditioning method, and it can be extended to random $n$-lifts of $G$, for any fixed $d$-regular graph $G$.