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Now let $B(G,H)$ = $min_{f}\\{ dilation(f) \\}$, over all such maps $f$.\n  The parameter $B(G,H)$ is a generalization of the classic and well studied \"bandwidth\" of $G$, defined as $B(G,P(n))$, where $P(n)$ is the path on $n$ points and $n = |V(G)|$. Let $[a_{1}\\times a_{2}\\times \\cdots \\times a_{k} ]$ be the $k$-dimensional grid graph with int"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1403.2749","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-03-11T20:45:51Z","cross_cats_sorted":[],"title_canon_sha256":"0f5e13a27d683c9aaa9a39051e0065ebae46fd40214f340595f277d757ad8ab9","abstract_canon_sha256":"41e4cbcd9e787569f5713568f778e7a252753db543230c3c1a94871bd5361417"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:34.245935Z","signature_b64":"dfGkiWNTHDWRtl0q+CNZ08ZXQKc2U/RrnmGV+wCv/+E8EBCHbugGysLGEd49hVhGFEDWbMlz/3AdDOmX9WD7AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4d4562dd05e1fbde47f8fed5cfed6ba9d6ad6537666f95cc731c5856614721b","last_reissued_at":"2026-05-18T02:56:34.245127Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:34.245127Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Embedding multidimensional grids into optimal hypercubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dan Pritikin, I.H. Sudborough, Zevi Miller","submitted_at":"2014-03-11T20:45:51Z","abstract_excerpt":"Let $G$ and $H$ be graphs, with $|V(H)|\\geq |V(G)| $, and $f:V(G)\\rightarrow V(H)$ a one to one map of their vertices. Let $dilation(f) = max\\{ dist_{H}(f(x),f(y)): xy\\in E(G) \\}$, where $dist_{H}(v,w)$ is the distance between vertices $v$ and $w$ of $H$. Now let $B(G,H)$ = $min_{f}\\{ dilation(f) \\}$, over all such maps $f$.\n  The parameter $B(G,H)$ is a generalization of the classic and well studied \"bandwidth\" of $G$, defined as $B(G,P(n))$, where $P(n)$ is the path on $n$ points and $n = |V(G)|$. 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