{"paper":{"title":"Minimal Sum Labeling of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Jakub Pek\\'arek, Jana Novotn\\'a, Martin T\\\"opfer, Mat\\v{e}j Kone\\v{c}n\\'y, Stanislav Ku\\v{c}era, \\v{S}t\\v{e}p\\'an \\v{S}imsa","submitted_at":"2017-08-01T23:52:46Z","abstract_excerpt":"A graph $G$ is called a sum graph if there is a so-called sum labeling of $G$, i.e. an injective function $\\ell: V(G) \\rightarrow \\mathbb{N}$ such that for every $u,v\\in V(G)$ it holds that $uv\\in E(G)$ if and only if there exists a vertex $w\\in V(G)$ such that $\\ell(u)+\\ell(v) = \\ell(w)$. We say that sum labeling $\\ell$ is minimal if there is a vertex $u\\in V(G)$ such that $\\ell(u)=1$. In this paper, we show that if we relax the conditions (either allow non-injective labelings or consider graphs with loops) then there are sum graphs without a minimal labeling, which partially answers the ques"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.00552","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"}