{"paper":{"title":"Edge Decompositions of Hypercubes by Paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Anick, Mark Ramras","submitted_at":"2013-08-22T18:37:08Z","abstract_excerpt":"Many authors have investigated edge decompositions of graphs by the edge sets of isomorphic copies of special subgraphs. For $q$- dimensional hypercubes $Q_q$ various researchers have done this for cer- tain trees, paths, and cycles. In this paper we shall say that \"$H$ divides $G$\" if $E(G)$ is the disjoint union of $\\{fE(H_i) | H_i \\simeq H\\}$. Our main result is that for $q$ odd and $q < 2^{32}$, the path of length $m$, $P_m$, divides $Q_q$ if and only if $m \\leq q$ and $m | (q \\times 2^{q-1})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4949","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"}