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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| Published in: | Theoretical computer science Vol. 941; pp. 278 - 299 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier B.V
04.01.2023
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| Subjects: | |
| ISSN: | 0304-3975, 1879-2294 |
| Online Access: | Get full text |
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| Summary: | 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. |
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| ISSN: | 0304-3975 1879-2294 |
| DOI: | 10.1016/j.tcs.2022.11.020 |