{"paper":{"title":"A Brooks type theorem for the maximum local edge connectivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bjarne Toft, Michael Stiebitz","submitted_at":"2016-03-30T13:42:27Z","abstract_excerpt":"For a graph $G$, let $\\cn(G)$ and $\\la(G)$ denote the chromatic number of $G$ and the maximum local edge connectivity of $G$, respectively. A result of Dirac \\cite{Dirac53} implies that every graph $G$ satisfies $\\cn(G)\\leq \\la(G)+1$. In this paper we characterize the graphs $G$ for which $\\cn(G)=\\la(G)+1$. The case $\\la(G)=3$ was already solved by Alboulker {\\em et al.\\,} \\cite{AlboukerV2016}. We show that a graph $G$ with $\\la(G)=k\\geq 4$ satisfies $\\cn(G)=k+1$ if and only if $G$ contains a block which can be obtained from copies of $K_{k+1}$ by repeated applications of the Haj\\'os join."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.09187","kind":"arxiv","version":1},"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"}