{"paper":{"title":"Distant total sum distinguishing index of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jakub Przyby{\\l}o","submitted_at":"2017-03-16T15:28:03Z","abstract_excerpt":"Let $c:V\\cup E\\to\\{1,2,\\ldots,k\\}$ be a proper total colouring of a graph $G=(V,E)$ with maximum degree $\\Delta$. We say vertices $u,v\\in V$ are sum distinguished if $c(u)+\\sum_{e\\ni u}c(e)\\neq c(v)+\\sum_{e\\ni v}c(e)$. By $\\chi\"_{\\Sigma,r}(G)$ we denote the least integer $k$ admitting such a colouring $c$ for which every $u,v\\in V$, $u\\neq v$, at distance at most $r$ from each other are sum distinguished in $G$. For every positive integer $r$ an infinite family of examples is known with $\\chi\"_{\\Sigma,r}(G)=\\Omega(\\Delta^{r-1})$. In this paper we prove that $\\chi\"_{\\Sigma,r}(G)\\leq (2+o(1))\\De"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05672","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"}