On the phase transitions of graph coloring and independent sets

TL;DR

This paper explores how phase transitions in graph coloring and independent set problems relate to acyclic orientations, revealing empirical peaks in certain orientations that correspond to problem hardness and easier instances.

Contribution

It introduces a novel empirical analysis linking acyclic orientations to phase transitions in graph coloring and independent sets, highlighting their role in problem difficulty.

Findings

Peak in 'shortest' acyclic orientations at chromatic number increase

Maxima in 'widest' orientations coincide with independence number changes

Empirical evidence of phase transition phenomena in random graphs

Abstract

We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph's chromatic number and independence number. We provide empirical evidence that, along a sequence of increasingly denser random graphs, the fraction of acyclic orientations that are `shortest' peaks when the chromatic number increases, and that such maxima tend to coincide with locally easiest instances of the problem. Similar evidence is provided concerning the `widest' acyclic orientations and the independence number.