{"paper":{"title":"Linear and cyclic distance-three labellings of trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Deborah King, Kelvin Yang Li, Sanming Zhou","submitted_at":"2013-09-06T06:33:59Z","abstract_excerpt":"Given a finite or infinite graph $G$ and positive integers $\\ell, h_1, h_2, h_3$, an $L(h_1, h_2, h_3)$-labelling of $G$ with span $\\ell$ is a mapping $f: V(G) \\rightarrow \\{0, 1, 2, \\ldots, \\ell\\}$ such that, for $i = 1, 2, 3$ and any $u, v \\in V(G)$ at distance $i$ in $G$, $|f(u) - f(v)| \\geq h_i$. A $C(h_1, h_2, h_3)$-labelling of $G$ with span $\\ell$ is defined similarly by requiring $|f(u) - f(v)|_{\\ell} \\ge h_i$ instead, where $|x|_{\\ell} = \\min\\{|x|, \\ell-|x|\\}$. The minimum span of an $L(h_1, h_2, h_3)$-labelling, or a $C(h_1, h_2, h_3)$-labelling, of $G$ is denoted by $\\lambda_{h_1,h_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.1545","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"}