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In particular, if $G$ is a triangle and $K_{1,r}$-free graph, then for any graph $H$, $\\gamma(G\\square H)\\geq \\frac{r}{2r-1}\\gamma(G)\\gamma(H)$. If $G$ is a $K_r$ and $P_5$-free graph for some integer $r\\geq 2$, then for any graph $H$, $\\gamma(G\\square H)\\geq \\frac{r-1}{2r-3}\\gamma(G)\\gamma(H)$. We do this by bounding the power of $G$, $\\pi(G)$. We show that if $G$ is claw"},"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":"1705.04954","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-14T11:51:46Z","cross_cats_sorted":[],"title_canon_sha256":"0ea59d591f117e3dea28e69714ab16f77155418ee0cecf9a31d3ed22cfb48a15","abstract_canon_sha256":"8ba455e33fad0674969177721470fbf6c270a5e844bc205fbb1628638c6d7d15"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:34.238486Z","signature_b64":"yc48NKifOTQwsYGHcq8VC8cWU9euQopQ1jWEE89VmNqm970q5eq643v0cCgGKEQqT3fsxn8hAQKvmMokldaNCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb37872c1a387395d044a8d676747e93effbc082f48f586f8ea333fe8a13841b","last_reissued_at":"2026-05-18T00:44:34.238064Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:34.238064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vizing-type bounds for graphs with induced subgraph restrictions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Elliot Krop, Gaspar Porta, Pritul Patel","submitted_at":"2017-05-14T11:51:46Z","abstract_excerpt":"For any graphs $G$ and $H$, we say that a bound is of Vizing-type if $\\gamma(G\\square H)\\geq c \\gamma(G)\\gamma(H)$ for some constant $c$. We show several bounds of Vizing-type for graphs $G$ with forbidden induced subgraphs. In particular, if $G$ is a triangle and $K_{1,r}$-free graph, then for any graph $H$, $\\gamma(G\\square H)\\geq \\frac{r}{2r-1}\\gamma(G)\\gamma(H)$. If $G$ is a $K_r$ and $P_5$-free graph for some integer $r\\geq 2$, then for any graph $H$, $\\gamma(G\\square H)\\geq \\frac{r-1}{2r-3}\\gamma(G)\\gamma(H)$. We do this by bounding the power of $G$, $\\pi(G)$. 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