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Vizing conjectured that $\\gamma(G \\square H) \\geq \\gamma(G)\\gamma(H)$ where $\\square$ stands for the Cartesian product of graphs. In this note, we define classes of graphs $\\mathcal{A}_n$, for $n\\geq 0$, so that every graph belongs to some such class, and $\\mathcal{A}_0$ corresponds to class $A$ of Bartsalkin and German. We prove that "},"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":"1512.01077","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-03T13:59:56Z","cross_cats_sorted":[],"title_canon_sha256":"ffb5cf713956edad6ad44da9c16b0ebf0c6de196bd7938c9d373340bfaaf0d3d","abstract_canon_sha256":"85df34859428b6fc9c7fa277dbe9aaec6944ade0b225ba540807f4a40ce2c34c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:46.891222Z","signature_b64":"v1dv7nTIRuLxC3g+FqWiBO/qMK8M+u+Bc0lTbZ9CNCm4Ssm6oES8O0kKQ9zDcCRRSapVJZQWYHb8iL1aj/QkDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e38eb1508d12c3467251da4748f3b759b01afd52eb036bf7344f2de14836ea1d","last_reissued_at":"2026-05-18T01:17:46.890412Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:46.890412Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A class of graphs approaching Vizing's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aziz Contractor, Elliot Krop","submitted_at":"2015-12-03T13:59:56Z","abstract_excerpt":"For any graph $G=(V,E)$, a subset $S\\subseteq V$ \\emph{dominates} $G$ if all vertices are contained in the closed neighborhood of $S$, that is $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written $\\gamma(G)$. In 1963, V.G. Vizing conjectured that $\\gamma(G \\square H) \\geq \\gamma(G)\\gamma(H)$ where $\\square$ stands for the Cartesian product of graphs. In this note, we define classes of graphs $\\mathcal{A}_n$, for $n\\geq 0$, so that every graph belongs to some such class, and $\\mathcal{A}_0$ corresponds to class $A$ of Bartsalkin and German. 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