{"paper":{"title":"Distant sum distinguishing index of graphs with bounded minimum degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jakub Przyby{\\l}o","submitted_at":"2017-03-14T23:06:57Z","abstract_excerpt":"For any graph $G=(V,E)$ with maximum degree $\\Delta$ and without isolated edges, and a positive integer $r$, by $\\chi'_{\\Sigma,r}(G)$ we denote the $r$-distant sum distinguishing index of $G$. This is the least integer $k$ for which a proper edge colouring $c:E\\to\\{1,2,\\ldots,k\\}$ exists such that $\\sum_{e\\ni u}c(e)\\neq \\sum_{e\\ni v}c(e)$ for every pair of distinct vertices $u,v$ at distance at most $r$ in $G$. It was conjectured that $\\chi'_{\\Sigma,r}(G)\\leq (1+o(1))\\Delta^{r-1}$ for every $r\\geq 3$. Thus far it has been in particular proved that $\\chi'_{\\Sigma,r}(G)\\leq 6\\Delta^{r-1}$ if $r\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.04815","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"}