{"paper":{"title":"Improved bounds on coloring of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aldo Procacci, Benedetto Scoppola, Sokol Ndreca","submitted_at":"2010-05-11T16:05:46Z","abstract_excerpt":"Given a graph $G$ with maximum degree $\\Delta\\ge 3$, we prove that the acyclic edge chromatic number $a'(G)$ of $G$ is such that $a'(G)\\le\\lceil 9.62 (\\Delta-1)\\rceil$. Moreover we prove that:\n  $a'(G)\\le \\lceil 6.42(\\Delta-1)\\rceil$ if $G$ has girth $g\\ge 5\\,$; $a'(G)\\le \\lceil5.77 (\\Delta-1)\\rc$ if\n  $G$ has girth $g\\ge 7$; $a'(G)\\le \\lc4.52(\\D-1)\\rc$ if $g\\ge 53$;\n  $a'(G)\\le \\D+2\\,$ if $g\\ge \\lceil25.84\\D\\log\\D(1+ 4.1/\\log\\D)\\rceil$.\n  We further prove that the acyclic (vertex) chromatic number $a(G)$ of $G$ is such that\n  $a(G)\\le \\lc 6.59 \\Delta^{4/3}+3.3\\D\\rc$. We also prove that the st"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.1875","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"}