There are finitely many $5$-vertex-critical $(P_6,\text{bull})$-free graphs
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| Format: | Preprint |
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2025
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| _version_ | 1866915251335725056 |
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| author | Ju, Yiao Jooken, Jorik Goedgebeur, Jan Huang, Shenwei |
| author_facet | Ju, Yiao Jooken, Jorik Goedgebeur, Jan Huang, Shenwei |
| contents | In this paper, we are interested in $4$-colouring algorithms for graphs that do not contain an induced path on $6$ vertices nor an induced bull, i.e., the graph with vertex set $\{v_1,v_2,v_3,v_4,v_5\}$ and edge set $\{v_1v_2,v_2v_3,v_3v_4,v_2v_5,v_3v_5\}$. Such graphs are referred to as $(P_6,\text{bull})$-free graphs. A graph $G$ is \emph{$k$-vertex-critical} if $χ(G)=k$, and every proper induced subgraph $H$ of $G$ has $χ(H)<k$. In the current paper, we investigate the structure of $5$-vertex-critical $(P_6,\text{bull})$-free graphs and show that there are only finitely many such graphs, thereby answering a question of Maffray and Pastor. A direct corollary of this is that there exists a polynomial-time algorithm to decide if a $(P_6,\text{bull})$-free graph is $4$-colourable such that this algorithm can also provide a certificate that can be verified in polynomial time and serves as a proof of 4-colourability or non-4-colourability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_14134 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | There are finitely many $5$-vertex-critical $(P_6,\text{bull})$-free graphs Ju, Yiao Jooken, Jorik Goedgebeur, Jan Huang, Shenwei Combinatorics In this paper, we are interested in $4$-colouring algorithms for graphs that do not contain an induced path on $6$ vertices nor an induced bull, i.e., the graph with vertex set $\{v_1,v_2,v_3,v_4,v_5\}$ and edge set $\{v_1v_2,v_2v_3,v_3v_4,v_2v_5,v_3v_5\}$. Such graphs are referred to as $(P_6,\text{bull})$-free graphs. A graph $G$ is \emph{$k$-vertex-critical} if $χ(G)=k$, and every proper induced subgraph $H$ of $G$ has $χ(H)<k$. In the current paper, we investigate the structure of $5$-vertex-critical $(P_6,\text{bull})$-free graphs and show that there are only finitely many such graphs, thereby answering a question of Maffray and Pastor. A direct corollary of this is that there exists a polynomial-time algorithm to decide if a $(P_6,\text{bull})$-free graph is $4$-colourable such that this algorithm can also provide a certificate that can be verified in polynomial time and serves as a proof of 4-colourability or non-4-colourability. |
| title | There are finitely many $5$-vertex-critical $(P_6,\text{bull})$-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.14134 |