{"paper":{"title":"A Brooks-type result for sparse critical graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Kostochka, Matthew Yancey","submitted_at":"2014-08-05T01:12:33Z","abstract_excerpt":"A graph $G$ is $k$-{\\em critical} if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. Recently the authors gave a lower bound, $f_k(n) \\geq \\left\\lceil \\frac{(k+1)(k-2)|V(G)|-k(k-3)}{2(k-1)}\\right\\rceil$, that solves a conjecture by Gallai from 1963 and is sharp for every $n\\equiv 1\\,({\\rm mod }\\, k-1)$. It is also sharp for $k=4$ and every $n\\geq 6$. In this paper we refine the result by describing all $n$-vertex $k$-critical graphs $G$ with $|E(G)|= \\frac{(k+1)(k-2)|V(G)|-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.0846","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"}