Journal Articles Theory of Computing Year : 2024

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 ≥ 3 and ε ≥ 0, a randomised polynomial-time algorithm for colouring graphs of maximum degree  ∆ in which each vertex is contained in at most t copies of a cycle of length k, where 1 ≤ t ≤ ∆^(2ε / (1+2ε)) / (log ∆)² with (1 + ε)·∆ / log(∆/ √ t) colours. This unifies, 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). The bound on the chromatic number is tight up to an asymptotic factor 2 due to the random regular graph, and as such our work coincides with a famous algorithmic barrier to colouring random graphs, greatly expanding the range of known algorithms for graph colouring that meet this barrier.
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Dates and versions

hal-02548191 , version 1 (20-04-2020)
hal-02548191 , version 2 (19-11-2024)

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Ewan Davies, Ross Kang, François Pirot, Jean-Sébastien Sereni. An algorithmic framework for colouring locally sparse graphs. Theory of Computing, inPress. ⟨hal-02548191v2⟩
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