Some Results on Critical ($P_5,H$)-free Graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913258564222976 |
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| author | Xia, Wen Jooken, Jorik Goedgebeur, Jan Huang, Shenwei |
| author_facet | Xia, Wen Jooken, Jorik Goedgebeur, Jan Huang, Shenwei |
| contents | Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$, but $G$ has chromatic number $k$. The study of $k$-vertex-critical graphs for specific graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there exists a polynomial-time certifying algorithm to decide the $k$-colorability of a graph in the class. In this paper, we show that: (1) for $k \ge 1$, there are finitely many $k$-vertex-critical $(P_5,K_{1,4}+P_1)$-free graphs; (2) for $s \ge 1$, there are finitely many 5-vertex-critical $(P_5,K_{1,s}+P_1)$-free graphs; (3) for $k \ge 1$, there are finitely many $k$-vertex-critical $(P_5,\overline{K_3+2P_1})$-free graphs. Moreover, we characterize all $5$-vertex-critical $(P_5,H)$-free graphs where $H \in \{K_{1,3}+P_1,K_{1,4}+P_1,\overline{K_3+2P_1}\}$ using an exhaustive graph generation algorithm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05611 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some Results on Critical ($P_5,H$)-free Graphs Xia, Wen Jooken, Jorik Goedgebeur, Jan Huang, Shenwei Combinatorics Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$, but $G$ has chromatic number $k$. The study of $k$-vertex-critical graphs for specific graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there exists a polynomial-time certifying algorithm to decide the $k$-colorability of a graph in the class. In this paper, we show that: (1) for $k \ge 1$, there are finitely many $k$-vertex-critical $(P_5,K_{1,4}+P_1)$-free graphs; (2) for $s \ge 1$, there are finitely many 5-vertex-critical $(P_5,K_{1,s}+P_1)$-free graphs; (3) for $k \ge 1$, there are finitely many $k$-vertex-critical $(P_5,\overline{K_3+2P_1})$-free graphs. Moreover, we characterize all $5$-vertex-critical $(P_5,H)$-free graphs where $H \in \{K_{1,3}+P_1,K_{1,4}+P_1,\overline{K_3+2P_1}\}$ using an exhaustive graph generation algorithm. |
| title | Some Results on Critical ($P_5,H$)-free Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.05611 |