{"paper":{"title":"Counting critical subgraphs in $k$-critical graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Ma, Tianchi Yang","submitted_at":"2019-06-23T15:44:56Z","abstract_excerpt":"Gallai asked in 1984 if any $k$-critical graph on $n$ vertices contains at least $n$ distinct $(k-1)$-critical subgraphs. The answer is trivial for $k\\leq 3$. Improving a result of Stiebitz, Abbott and Zhou proved in 1995 that for all $k\\geq 4$, such graph contains $\\Omega(n^{1/(k-1)})$ distinct $(k-1)$-critical subgraphs. Since then no progress had been made until very recently, Hare resolved the case $k=4$ by showing that any $4$-critical graph on $n$ vertices contains at least $(8n-29)/3$ odd cycles.\n  In this paper, we mainly focus on 4-critical graphs and develop some novel tools for coun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.09598","kind":"arxiv","version":2},"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"}