{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:5M3YOLA2HBZZLUCEVDLHM5D6SP","short_pith_number":"pith:5M3YOLA2","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"},"canonical_sha256":"eb37872c1a387395d044a8d676747e93effbc082f48f586f8ea333fe8a13841b","source":{"kind":"arxiv","id":"1705.04954","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.04954","created_at":"2026-05-18T00:44:34Z"},{"alias_kind":"arxiv_version","alias_value":"1705.04954v1","created_at":"2026-05-18T00:44:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.04954","created_at":"2026-05-18T00:44:34Z"},{"alias_kind":"pith_short_12","alias_value":"5M3YOLA2HBZZ","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"5M3YOLA2HBZZLUCE","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"5M3YOLA2","created_at":"2026-05-18T12:31:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:5M3YOLA2HBZZLUCEVDLHM5D6SP","target":"record","payload":{"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"},"canonical_sha256":"eb37872c1a387395d044a8d676747e93effbc082f48f586f8ea333fe8a13841b","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"},"source_kind":"arxiv","source_id":"1705.04954","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:44:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"adanmdHjsaXb8+4wukH2Udv92VE3/I5Em80VMNz/oeYbuf26WTegkJ0Cg0pFqqZrTDJPHJ0rdk+VsBVpHgBWBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T15:36:24.879068Z"},"content_sha256":"bd6d1f7492a0642abde71367e21fbf79fd0368103507dc059af70fd6b997aaba","schema_version":"1.0","event_id":"sha256:bd6d1f7492a0642abde71367e21fbf79fd0368103507dc059af70fd6b997aaba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:5M3YOLA2HBZZLUCEVDLHM5D6SP","target":"graph","payload":{"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 $G$, $\\pi(G)$. We show that if $G$ is claw"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04954","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:44:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tL8SUGOmZmOwY9wNcfl8A9TJvyHMl7hwFFENzfo90liGHOLJe+xbykq83Yzai9lp6hgl0CvJgRbl4q/K3A9FBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T15:36:24.879431Z"},"content_sha256":"23666031f089876242aa27e9a098c17bea8a4d47e4d13d6dec690808ec1b73b8","schema_version":"1.0","event_id":"sha256:23666031f089876242aa27e9a098c17bea8a4d47e4d13d6dec690808ec1b73b8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5M3YOLA2HBZZLUCEVDLHM5D6SP/bundle.json","state_url":"https://pith.science/pith/5M3YOLA2HBZZLUCEVDLHM5D6SP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5M3YOLA2HBZZLUCEVDLHM5D6SP/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-01T15:36:24Z","links":{"resolver":"https://pith.science/pith/5M3YOLA2HBZZLUCEVDLHM5D6SP","bundle":"https://pith.science/pith/5M3YOLA2HBZZLUCEVDLHM5D6SP/bundle.json","state":"https://pith.science/pith/5M3YOLA2HBZZLUCEVDLHM5D6SP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5M3YOLA2HBZZLUCEVDLHM5D6SP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:5M3YOLA2HBZZLUCEVDLHM5D6SP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8ba455e33fad0674969177721470fbf6c270a5e844bc205fbb1628638c6d7d15","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-14T11:51:46Z","title_canon_sha256":"0ea59d591f117e3dea28e69714ab16f77155418ee0cecf9a31d3ed22cfb48a15"},"schema_version":"1.0","source":{"id":"1705.04954","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.04954","created_at":"2026-05-18T00:44:34Z"},{"alias_kind":"arxiv_version","alias_value":"1705.04954v1","created_at":"2026-05-18T00:44:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.04954","created_at":"2026-05-18T00:44:34Z"},{"alias_kind":"pith_short_12","alias_value":"5M3YOLA2HBZZ","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"5M3YOLA2HBZZLUCE","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"5M3YOLA2","created_at":"2026-05-18T12:31:00Z"}],"graph_snapshots":[{"event_id":"sha256:23666031f089876242aa27e9a098c17bea8a4d47e4d13d6dec690808ec1b73b8","target":"graph","created_at":"2026-05-18T00:44:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $G$, $\\pi(G)$. We show that if $G$ is claw","authors_text":"Elliot Krop, Gaspar Porta, Pritul Patel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-14T11:51:46Z","title":"Vizing-type bounds for graphs with induced subgraph restrictions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04954","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:bd6d1f7492a0642abde71367e21fbf79fd0368103507dc059af70fd6b997aaba","target":"record","created_at":"2026-05-18T00:44:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8ba455e33fad0674969177721470fbf6c270a5e844bc205fbb1628638c6d7d15","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-14T11:51:46Z","title_canon_sha256":"0ea59d591f117e3dea28e69714ab16f77155418ee0cecf9a31d3ed22cfb48a15"},"schema_version":"1.0","source":{"id":"1705.04954","kind":"arxiv","version":1}},"canonical_sha256":"eb37872c1a387395d044a8d676747e93effbc082f48f586f8ea333fe8a13841b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eb37872c1a387395d044a8d676747e93effbc082f48f586f8ea333fe8a13841b","first_computed_at":"2026-05-18T00:44:34.238064Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:34.238064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yc48NKifOTQwsYGHcq8VC8cWU9euQopQ1jWEE89VmNqm970q5eq643v0cCgGKEQqT3fsxn8hAQKvmMokldaNCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:34.238486Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.04954","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bd6d1f7492a0642abde71367e21fbf79fd0368103507dc059af70fd6b997aaba","sha256:23666031f089876242aa27e9a098c17bea8a4d47e4d13d6dec690808ec1b73b8"],"state_sha256":"73d55fc129000005a0a25f0334e2eb7dda6f99d5a96fbd00f515d673e7695539"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J/uooSVqc1oZLavRB4RBZq4omhijw+UFYIzbDbw01esDha+4OyMZVN+K2vBU8TPGlLuuYcaBzuRolc8ZXyLvAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T15:36:24.881455Z","bundle_sha256":"83dbcc21c3503290981584b8df2497521be8297546119a8fffce8a332a0c4359"}}