{"paper":{"title":"Beyond the Vizing's bound for at most seven colors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"cs.DS","authors_text":"{\\L}ukasz Kowalik, Marcin Kami\\'nski","submitted_at":"2012-11-21T13:52:26Z","abstract_excerpt":"Let $G=(V,E)$ be a simple graph of maximum degree $\\Delta$. The edges of $G$ can be colored with at most $\\Delta +1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\\Delta$ colors.\n  Vizing's Theorem gives a bound of $\\frac{\\Delta}{\\Delta+1}|E|$. This is known to be tight for cliques $K_{\\Delta+1}$ when $\\Delta$ is even. However, for $\\Delta=3$ it was improved to $26/31|E|$ by Albertson and Haas [Parsimonious edge colorings, Disc. Math. 148, 1996] and later to $6/7|E|$ by Rizzi [Approximating the maximum 3-edge-colorable subgraph prob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5031","kind":"arxiv","version":4},"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"}