An algorithmic framework for colouring locally sparse graphs

TL;DR

This paper introduces a new algorithmic framework for graph colouring that leverages local probabilistic properties, enabling efficient colouring of certain sparse graphs and extending previous theoretical bounds.

Contribution

It presents a novel probabilistic framework for graph colouring that generalizes and improves existing bounds for locally sparse graphs, matching known algorithmic barriers.

Findings

Provides a polynomial-time randomized colouring algorithm for graphs with bounded cycle structures.

Achieves bounds on chromatic number that are tight up to a factor of two.

Extends previous results by Kim, Alon et al., Molloy, and others.

Abstract

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $\Delta$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2\le t\le \Delta^\frac{2\varepsilon}{1+2\varepsilon}/(\log\Delta)^2$, with $\lfloor(1+\varepsilon)\Delta/\log(\Delta/\sqrt t)\rfloor$ colours. This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor $2$ and it coincides with a famous algorithmic barrier to colouring random graphs.