Critical Exponent for the Acyclic Chromatic Number of Random Graphs

TL;DR

This paper investigates the phase transition of the acyclic chromatic number in random graphs, identifying a critical exponent for the growth rate and analyzing a relaxed version allowing some cycles.

Contribution

It introduces a phase transition analysis for the acyclic chromatic number in random graphs and establishes the critical exponent for both strict and relaxed conditions.

Findings

Acyclic chromatic number transitions from sublinear to linear growth as edge probability increases.

The critical exponent is zero when a small fraction of cycles are allowed to violate acyclicity.

The phase transition occurs even in sparse random graphs.

Abstract

In this paper we study acyclic colouring in the random subgraph $\mathit{G}$ of the complete graph $\mathit{K}_n$ on $\mathit{n}$ vertices where each edge is present with probability $\mathit{p}$; independent of the other edges. We show that the acyclic chromatic number exhibits a phase transition from sublinear to linear growth as the edge probability increases, even in the sparse regime and obtain estimates for the critical exponent. Next, we introduce a relaxation by allowing for a small fraction of "bad" cycles to violate the acyclic colouring condition and show that the critical exponent in this case is in fact zero, no matter how small the fraction.