{"paper":{"title":"$k$-Distance Magic Labeling and Long Brush Graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"V. Vilfred Kamalappan","submitted_at":"2022-11-13T17:49:19Z","abstract_excerpt":"We define a labeling $f:$ $V(G)$ $\\rightarrow$ $\\{1, 2, \\ldots, n\\}$ on a graph $G$ of order $n \\geq 3$ as a \\emph{$k$-distance magic} ($k$-DM) if $\\sum_{w\\in \\partial N_k(u)}{ f(w)}$ is a constant and independent of $u\\in V(G)$ where $\\partial N_k(u)$ = $\\{v\\in V(G): d(u, v) = k\\}$, $k\\in\\mathbb{N}$. Graph $G$ is called a \\emph{$k$-DM} if it has a $k$-DM labeling(L). Long Brush is a graph $G$ with $V(G)$ = $\\{u_1, u_2, . . . , u_n,$ $v_1, v_2, . . . , v_{m}\\}$, a path $P_n$ = $u_1$ $u_2$ . . . $u_n$ and $E(G)$ = $E(P_n)$ $\\cup$ $\\{u_1v_i:$ $i$ = 1 to $m\\}$ $\\cup$ $E(<v_1, v_2, . . . , v_{m}>)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.09666","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.09666/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}