Dr Barnaby Martin barnaby.d.martin@durham.ac.uk
Associate Professor
Colouring graphs of bounded diameter in the absence of small cycles
Martin, B.; Paulusma, D.; Smith, S.
Authors
Professor Daniel Paulusma daniel.paulusma@durham.ac.uk
Professor
Siani Alice Smith siani.smith@durham.ac.uk
PGR Student Doctor of Philosophy
Abstract
For k ≥ 1, a k-colouring c of G is a mapping from V (G) to {1, 2, . . . , k} such that c(u) 6= c(v) for any two adjacent vertices u and v. The k-Colouring problem is to decide if a graph G has a k-colouring. For a family of graphs H, a graph G is H-free if G does not contain any graph from H as an induced subgraph. Let Cs be the s-vertex cycle. In previous work (MFCS 2019) we examined the effect of bounding the diameter on the complexity of 3-Colouring for (C3, . . . , Cs)-free graphs and H-free graphs where H is some polyad. Here, we prove for certain small values of s that 3-Colouring is polynomial-time solvable for Cs-free graphs of diameter 2 and (C4, Cs)-free graphs of diameter 2. In fact, our results hold for the more general problem List 3-Colouring. We complement these results with some hardness result for diameter 4.
Citation
Martin, B., Paulusma, D., & Smith, S. (2022). Colouring graphs of bounded diameter in the absence of small cycles. Discrete Applied Mathematics, 314, 150-161. https://doi.org/10.1016/j.dam.2022.02.026
Journal Article Type | Article |
---|---|
Acceptance Date | Feb 22, 2022 |
Online Publication Date | Mar 19, 2022 |
Publication Date | Jun 15, 2022 |
Deposit Date | May 18, 2022 |
Publicly Available Date | Mar 20, 2024 |
Journal | Discrete Applied Mathematics |
Print ISSN | 0166-218X |
Publisher | Elsevier |
Peer Reviewed | Peer Reviewed |
Volume | 314 |
Pages | 150-161 |
DOI | https://doi.org/10.1016/j.dam.2022.02.026 |
Public URL | https://durham-repository.worktribe.com/output/1206956 |
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Copyright Statement
© 2022. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/
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