A refinement on the structure of vertex-critical ($P_5$, gem)-free graphs
Abstract
We give a new, stronger proof that there are only finitely many $k$-vertex-critical ($P_5$,~gem)-free graphs for all $k$. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all $6$- and $7$-vertex-critical $(P_5$, gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the $k$-colourability of $(P_5$, gem)-free graphs for all $k$ where the certificate is either a $k$-colouring or a $(k+1)$-vertex-critical induced subgraph. Our complete lists for $k\le 7$ allow for the implementation of these algorithms for all $k\le 6$.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2022
- DOI:
- 10.48550/arXiv.2212.04659
- arXiv:
- arXiv:2212.04659
- Bibcode:
- 2022arXiv221204659C
- Keywords:
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- Mathematics - Combinatorics;
- Computer Science - Discrete Mathematics;
- Computer Science - Data Structures and Algorithms