{"paper":{"title":"Distant irregularity strength 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":"2017-03-08T11:05:35Z","abstract_excerpt":"Consider a graph $G=(V,E)$ without isolated edges and with maximum degree $\\Delta$. Given a colouring $c:E\\to\\{1,2,\\ldots,k\\}$, the weighted degree of a vertex $v\\in V$ is the sum of its incident colours, i.e., $\\sum_{e\\ni v}c(e)$. For any integer $r\\geq 2$, the least $k$ admitting the existence of such $c$ attributing distinct weighted degrees to any two different vertices at distance at most $r$ in $G$ is called the $r$-distant irregularity strength of $G$ and denoted by $s_r(G)$. This graph invariant provides a natural link between the well known 1--2--3 Conjecture and irregularity strength"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.02787","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"}