An algorithmic framework for colouring locally sparse graphs

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 $\epsilon >0$, 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 $1/2\le t\le {\Delta }^{\frac{2\epsilon }{1+2\epsilon }}/(\log \Delta {)}^2$, with $\lfloor (1+\epsilon )\Delta /\log (\Delta /\sqrt{t})\rfloor$ 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.

23 pages

References in corpus (6)

• Phase Transitions in the Coloring of Random Graphs
• Acyclic edge-coloring using entropy compression
• On independent sets in random graphs
• On the average size of independent sets in triangle-free graphs
• Colouring triangle-free graphs with local list sizes
• On the Lovász Theta function for Independent Sets in Sparse Graphs

Cited by in corpus (1)

• An improved procedure for colouring graphs of bounded local density