{"paper":{"title":"Exhaustive generation of $k$-critical $\\mathcal H$-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jan Goedgebeur, Oliver Schaudt","submitted_at":"2015-06-11T12:29:14Z","abstract_excerpt":"We describe an algorithm for generating all $k$-critical $\\mathcal H$-free graphs, based on a method of Ho\\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes.\n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03647","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}