{"paper":{"title":"Further results on the deficiency of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Hrant H. Khachatrian, Petros A. Petrosyan","submitted_at":"2016-08-02T17:08:14Z","abstract_excerpt":"A \\emph{proper $t$-edge-coloring} of a graph $G$ is a mapping $\\alpha: E(G)\\rightarrow \\{1,\\ldots,t\\}$ such that all colors are used, and $\\alpha(e)\\neq \\alpha(e^{\\prime})$ for every pair of adjacent edges $e,e^{\\prime}\\in E(G)$. If $\\alpha $ is a proper edge-coloring of a graph $G$ and $v\\in V(G)$, then \\emph{the spectrum of a vertex $v$}, denoted by $S\\left(v,\\alpha \\right)$, is the set of all colors appearing on edges incident to $v$. \\emph{The deficiency of $\\alpha$ at vertex $v\\in V(G)$}, denoted by $def(v,\\alpha)$, is the minimum number of integers which must be added to $S\\left(v,\\alpha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.00904","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"}