{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/critical-p-5-w-4-free-graphs","title":"Critical $(P_5,W_4)$-Free Graphs","arxiv_id":"2501.04923","date":"2025-01-09","proceeding":null,"authors":["Wen Xia","Jorik Jooken","Jan Goedgebeur","Iain Beaton","Ben Cameron","Shenwei Huang"],"abstract":"A graph $G$ is $k$-vertex-critical if $\\chi(G) = k$ but $\\chi(G-v)<k$ for all $v \\in V(G)$. A graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A $W_4$ is the graph consisting of a $C_4$ plus an additional vertex adjacent to all the vertices of the $C_4$. We show that there are finitely many $k$-vertex-critical $(P_5,W_4)$-free graphs for all $k \\ge 1$ and we characterize all $5$-vertex-critical $(P_5,W_4)$-free graphs. Our results imply the existence of a polynomial-time certifying algorithm to decide the $k$-colorability of $(P_5,W_4)$-free graphs for each $k \\ge 1$ where the certificate is either a $k$-coloring or a $(k+1)$-vertex-critical induced subgraph.","url_abs":"https://arxiv.org/abs/2501.04923v1","url_pdf":"https://arxiv.org/pdf/2501.04923v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"critical-p-5-w-4-free-graphs","repo_url":"https://github.com/jorikjooken/kvertexcriticalgraphs","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}