On the chromatic number of the preferential attachment graph

TL;DR

This paper establishes that the chromatic number of the preferential attachment graph $PA_t(m, \, \delta)$ converges to $m+1$ with high probability, for certain parameters, using combinatorial and probabilistic methods.

Contribution

It provides a rigorous proof that the chromatic number of these graphs asymptotically equals $m+1$, introducing a new combinatorial approach for this class of graphs.

Findings

Chromatic number asymptotically equals m+1

Construction of digraphs with chromatic number m+1

Almost sure embedding of these digraphs in the graph

Abstract

We prove that for every $m\in \mathbb N$ and every $\delta\in (-m,0)$, the chromatic number of the preferential attachment graph $PA_t(m, \delta)$ is asymptotically almost surely equal to $m+1$. The proof relies on a combinatorial construction of a family of digraphs of chromatic number $m+1$ followed by a proof that asymptotically almost surely there is a digraph in this family, which is realised as a subgraph of the preferential attachment graph.