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Examining this more carefully, consider the minimum size of a connected dominating set of vertices $\\gamma_c(Q_n)$, which is $2^n-L(Q_n)$ for $n\\ge2$. We show that $\\gamma_c(Q_n)\\sim 2^n/n$. We use Hamming codes and an \"expansion\" method to construct leafy spanning trees in $Q_n$."},"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":"1905.13292","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-30T20:36:47Z","cross_cats_sorted":[],"title_canon_sha256":"d5a174979ec3e5ddeb96c4efcb8514a9ca80198e965859a76f8c97a4b83dd2ad","abstract_canon_sha256":"1838225bea591d696afc67f21abc2dca3af8d1627ca455f4a78ee60ecb785f90"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:35.397709Z","signature_b64":"AOLHJWJGXhbjCA8uiMAQemyBo5oAtbInZqfNtjxstiOF7ijY82MjavbLOtAAtBG768BDQxdyQEOZO0WMJWZKDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0555432592bcd1d37809f59b8932c69103927a9c922d8639dd05942a7e2dd19f","last_reissued_at":"2026-05-17T23:44:35.397038Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:35.397038Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spanning Trees and Domination in Hypercubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jerrold R. Griggs","submitted_at":"2019-05-30T20:36:47Z","abstract_excerpt":"Let $L(G)$ denote the maximum number of leaves in any spanning tree of a connected graph $G$. We show the (known) result that for the $n$-cube $Q_n$, $L(Q_n) \\sim 2^n = |V(Q_n)|$ as $n\\rightarrow \\infty$. Examining this more carefully, consider the minimum size of a connected dominating set of vertices $\\gamma_c(Q_n)$, which is $2^n-L(Q_n)$ for $n\\ge2$. We show that $\\gamma_c(Q_n)\\sim 2^n/n$. 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