{"paper":{"title":"Distant total irregularity strength of graphs via random vertex ordering","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jakub Przyby{\\l}o","submitted_at":"2017-03-01T16:46:28Z","abstract_excerpt":"Let $c:V\\cup E\\to\\{1,2,\\ldots,k\\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\\Delta$. Two vertices $u,v\\in V$ are sum distinguished if they differ with respect to sums of their incident colours, i.e. $c(u)+\\sum_{e\\ni u}c(e)\\neq c(v)+\\sum_{e\\ni v}c(e)$. The least integer $k$ admitting such colouring $c$ under which every $u,v\\in V$ at distance $1\\leq d(u,v)\\leq r$ in $G$ are sum distinguished is denoted by ${\\rm ts}_r(G)$. Such graph invariants link the concept of the total vertex irregularity strength of graphs with so called 1-2-Conjecture, whose "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00376","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"}