Graph structure via local occupancy

TL;DR

This paper extends the understanding of local properties in graphs, using the hard-core model to guarantee large independent sets and strong colorings in broader classes of graphs beyond triangle-free, improving previous bounds and results.

Contribution

It introduces a unified framework based on local occupancy in the hard-core model to derive graph structure results for a wider range of graphs, including those with local sparsity.

Findings

Guarantees large independent sets in broader graph classes.

Ensures stronger coloring properties via Lovász local lemma.

Generalizes and improves previous bounds on graph structure.

Abstract

The first author together with Jenssen, Perkins and Roberts (2017) recently showed how local properties of the hard-core model on triangle-free graphs guarantee the existence of large independent sets, of size matching the best-known asymptotics due to Shearer (1983). The present work strengthens this in two ways: first, by guaranteeing stronger graph structure in terms of colourings through applications of the Lov\'asz local lemma; and second, by extending beyond triangle-free graphs in terms of local sparsity, treating for example graphs of bounded local edge density, of bounded local Hall ratio, and of bounded clique number. This generalises and improves upon much other earlier work, including that of Shearer (1995), Alon (1996) and Alon, Krivelevich and Sudakov (1999), and more recent results of Molloy (2019), Bernshteyn (2019) and Achlioptas, Iliopoulos and Sinclair (2019). Our results derive from a common framework built around the hard-core model. It pivots on a property we call local occupancy, giving a clean separation between the methods for deriving graph structure with probabilistic information and verifying the requisite probabilistic information itself.