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3-Colouring \(P_t\)-Free Graphs Without Short Odd Cycles

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Abstract

For any odd \(t\ge 9\), we present a polynomial-time algorithm that solves the 3-colouring problem, and finds a 3-colouring if one exists, in \(P_{t}\)-free graphs of odd girth at least \(t-2\). In particular, our algorithm works for \((P_9, C_3, C_5)\)-free graphs, thus making progress towards determining the complexity of 3-colouring in \(P_t\)-free graphs, which is open for \(t\ge 8\).

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Notes

  1. Because of Lemma 3.1, we need not worry about dominated vertices, and Claim 5.2 is the only place where we need to exclude them. In an algorithmic implementation of our method, one will ignore such vertices if detected by Claim 5.2, and colour them at the very end (in case a colouring is found).

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  1. Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile

    Alberto Rojas Anríquez & Maya Stein

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  1. Alberto Rojas Anríquez
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Supported by FONDECYT Regular Grant 1221905, by MathAmSud MATH190013, by FAPESP-CONICYT Investigación Conjunta grant Código 2019/13364-7, and by ANID Basal Grant CMM FB210005.

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Rojas Anríquez, A., Stein, M. 3-Colouring \(P_t\)-Free Graphs Without Short Odd Cycles. Algorithmica 85, 831–853 (2023). https://doi.org/10.1007/s00453-022-01049-0

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