{"paper":{"title":"Asymptotically optimal neighbor sum distinguishing total colorings of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jakub Przyby{\\l}o, Sarah Loeb, Yunfang Tang","submitted_at":"2015-07-27T20:17:17Z","abstract_excerpt":"Given a proper total $k$-coloring $c:V(G)\\cup E(G)\\to\\{1,2,\\ldots,k\\}$ of a graph $G$, we define the value of a vertex $v$ to be $c(v) + \\sum_{uv \\in E(G)} c(uv)$. The smallest integer $k$ such that $G$ has a proper total $k$-coloring whose values form a proper coloring is the neighbor sum distinguishing total chromatic number of $G$, $\\chi\"_{\\Sigma}(G)$. Pil\\'sniak and Wo\\'zniak (2013) conjectured that $\\chi\"_{\\Sigma}(G)\\leq \\Delta(G)+3$ for any simple graph with maximum degree $\\Delta(G)$. In this paper, we prove this bound to be asymptotically correct by showing that $\\chi\"_{\\Sigma}(G)\\leq "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07573","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"}