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It is conjectured that for all graphs $G$ and $H$, $f(G\\times H)\\leq f(G)f(H)$. If the graph $G$ satisfies the odd two-pebbling property, we will prove that $f(C_{4k+3}\\times G)\\leq f(C_{4k+3})f(G)$ and $f(M(C_{2n})\\times G)\\leq f(M(C_{2n}))f(G)$, where $C_{4k+3}$ is the odd cycle of order $4k+3$ and $M(C_{2n})$ is the middle gra"},"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":"1402.0764","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-04T15:33:12Z","cross_cats_sorted":[],"title_canon_sha256":"128b9575eefcc403c916ff753afe8a5e43ef02685598ff7f86c5e2f9b339b220","abstract_canon_sha256":"be632b1313c05f8c594df43089885ab302b2858d5eb1fa07af24e777f6466f12"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:00.058116Z","signature_b64":"sWpIeKI8RpMM1l5dbqWJ5HwUb/1vJ+ylaQ4ygjLp2bR09mx21Y9W33XwNKn5ujep/XZQtli0u15a0UpVOe5tCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85e0ce6732b5541a3e7eb2901356facbf8def3bbdc0865e59ff912f516be109a","last_reissued_at":"2026-05-18T03:00:00.057354Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:00.057354Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pebbling on $C_{4k+3}\\times G$ and $M(C_{2n})\\times G$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun-Ming Xu, Yong-Liang Pan, Zheng-Jiang Xia","submitted_at":"2014-02-04T15:33:12Z","abstract_excerpt":"The pebbling number of a graph $G$, $f(G)$, is the least $p$ such that, however $p$ pebbles are placed on the vertices of $G$, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. 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