{"paper":{"title":"Beyond the Shannon's Bound","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"cs.DS","authors_text":"Arkadiusz Soca{\\l}a, {\\L}ukasz Kowalik, Micha{\\l} Farnik","submitted_at":"2013-09-24T07:51:56Z","abstract_excerpt":"Let $G=(V,E)$ be a multigraph of maximum degree $\\Delta$. The edges of $G$ can be colored with at most $\\frac{3}{2}\\Delta$ colors by Shannon's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\\Delta$ colors.\n  Shannon's Theorem gives a bound of $\\frac{\\Delta}{\\lfloor\\frac{3}{2}\\Delta\\rfloor}|E|$. However, for $\\Delta=3$, Kami\\'{n}ski and Kowalik [SWAT'10] showed that there is a 3-edge-colorable subgraph of size at least $\\frac{7}{9}|E|$, unless $G$ has a connected component isomorphic to $K_3+e$ (a $K_3$ with an arbitrary edge doubled). Here we extend th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6069","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"}