Colouring graphs with no induced six-vertex path or diamond

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (P6, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (P6, diamond)-free graph G is no larger tha...

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Veröffentlicht in:Theoretical computer science Jg. 941; S. 278 - 299
Hauptverfasser: Goedgebeur, Jan, Huang, Shenwei, Ju, Yiao, Merkel, Owen
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier B.V 04.01.2023
Schlagworte:
Graph colouring
Lovász theta function
Polynomial-time algorithms
Strong perfect graph theorem
χ-boundedness
Graph colouring
Lovász theta function
Polynomial-time algorithms
Strong perfect graph theorem
χ-boundedness
ISSN:0304-3975, 1879-2294
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Zusammenfassung:The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (P6, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (P6, diamond)-free graph G is no larger than the maximum of 6 and the clique number of G. We do this by reducing the problem to imperfect (P6, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex-critical (P6, diamond, K6)-free graph. Together with the Lovász theta function, this gives a polynomial time algorithm to compute the chromatic number of (P6, diamond)-free graphs. •We obtain an improved bound on the chromatic number of (P6, diamond)-free graphs.•We show that there is one 6-vertex-critical (P6, diamond, K6)-free graph.•We give a polynomial-time algorithm to colour (P6, diamond)-free graphs optimally.
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2022.11.020