The t-improper chromatic number of random graphs

TL;DR

This paper investigates the asymptotic behaviour of the t-improper chromatic number in Erdős-Rényi random graphs, extending classical proper coloring results to cases where each color class induces a subgraph with bounded degree t.

Contribution

It provides a detailed analysis of how the t-improper chromatic number behaves asymptotically in G(n,p) for various growth rates of t(n), generalizing known results for proper coloring.

Findings

Characterizes the asymptotic behaviour of χ^t(G(n,p)) for constant p.

Describes the impact of t(n) growth on the chromatic number.

Extends classical proper coloring results to t-improper colorings.

Abstract

We consider the $t$-improper chromatic number of the Erd{\H o}s-R{\'e}nyi random graph $G(n,p)$. The t-improper chromatic number $\chi^t(G)$ of $G$ is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$, then this is the usual notion of proper colouring. When the edge probability $p$ is constant, we provide a detailed description of the asymptotic behaviour of $\chi^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$.