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In 2014, \\.{Z}ak proved that if $\\Delta (G_{1}), \\Delta (G_{2}) \\leq n-2$ and $|E_{1}| + |E_{2}| + \\max \\{ \\Delta (G_{1}), \\Delta (G_{2}) \\} \\leq 3n - 96n^{3/4} - 65$, then $G_{1}$ and $G_{2}$ pack. In the same paper, he conjectured that if $\\Delta (G_{1}), \\Delta (G_{2}) \\leq n-2$, then $|E_{1}| + |E_{2}| + \\max \\{ \\Delta (G_{1}), \\Delta (G_{2}) \\} \\leq 3n - 7$ is sufficient for $G_{1}$ and"},"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":"1508.03672","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-14T22:34:10Z","cross_cats_sorted":[],"title_canon_sha256":"cae7e8c2d8bb8015a1310bb303403c34425879067caf32901d640b04e1bcb8d3","abstract_canon_sha256":"1e8b72c373c5f2ad1a2f68c8fc053973cfd30da5783fc368fca8cd76b58295f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:15.463869Z","signature_b64":"2ijU8KOqhCfgTNkuCHkfchtNRkboawCefW7Vzj32VWHwLAeMBbGP8Bun0BRmfsWReq7nOiQj4f5SAswKDn13Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5119e9d92f50aaf249746d3c4ed3931848c7c1267f9b45424aa69f48a986fb8e","last_reissued_at":"2026-05-18T01:35:15.463490Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:15.463490Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Toward \\.Zak's conjecture on graph packing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Kostochka, Andrew McConvey, Derrek Yager, Ervin Gy\\H{o}ri","submitted_at":"2015-08-14T22:34:10Z","abstract_excerpt":"Two graphs $G_{1} = (V_{1}, E_{1})$ and $G_{2} = (V_{2}, E_{2})$, each of order $n$, pack if there exists a bijection $f$ from $V_{1}$ onto $V_{2}$ such that $uv \\in E_{1}$ implies $f(u)f(v) \\notin E_{2}$. In 2014, \\.{Z}ak proved that if $\\Delta (G_{1}), \\Delta (G_{2}) \\leq n-2$ and $|E_{1}| + |E_{2}| + \\max \\{ \\Delta (G_{1}), \\Delta (G_{2}) \\} \\leq 3n - 96n^{3/4} - 65$, then $G_{1}$ and $G_{2}$ pack. In the same paper, he conjectured that if $\\Delta (G_{1}), \\Delta (G_{2}) \\leq n-2$, then $|E_{1}| + |E_{2}| + \\max \\{ \\Delta (G_{1}), \\Delta (G_{2}) \\} \\leq 3n - 7$ is sufficient for $G_{1}$ and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.03672","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1508.03672","created_at":"2026-05-18T01:35:15.463548+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.03672v1","created_at":"2026-05-18T01:35:15.463548+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.03672","created_at":"2026-05-18T01:35:15.463548+00:00"},{"alias_kind":"pith_short_12","alias_value":"KEM6TWJPKCVP","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"KEM6TWJPKCVPESLU","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"KEM6TWJP","created_at":"2026-05-18T12:29:27.538025+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KEM6TWJPKCVPESLUNU6E5U4TDB","json":"https://pith.science/pith/KEM6TWJPKCVPESLUNU6E5U4TDB.json","graph_json":"https://pith.science/api/pith-number/KEM6TWJPKCVPESLUNU6E5U4TDB/graph.json","events_json":"https://pith.science/api/pith-number/KEM6TWJPKCVPESLUNU6E5U4TDB/events.json","paper":"https://pith.science/paper/KEM6TWJP"},"agent_actions":{"view_html":"https://pith.science/pith/KEM6TWJPKCVPESLUNU6E5U4TDB","download_json":"https://pith.science/pith/KEM6TWJPKCVPESLUNU6E5U4TDB.json","view_paper":"https://pith.science/paper/KEM6TWJP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.03672&json=true","fetch_graph":"https://pith.science/api/pith-number/KEM6TWJPKCVPESLUNU6E5U4TDB/graph.json","fetch_events":"https://pith.science/api/pith-number/KEM6TWJPKCVPESLUNU6E5U4TDB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KEM6TWJPKCVPESLUNU6E5U4TDB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KEM6TWJPKCVPESLUNU6E5U4TDB/action/storage_attestation","attest_author":"https://pith.science/pith/KEM6TWJPKCVPESLUNU6E5U4TDB/action/author_attestation","sign_citation":"https://pith.science/pith/KEM6TWJPKCVPESLUNU6E5U4TDB/action/citation_signature","submit_replication":"https://pith.science/pith/KEM6TWJPKCVPESLUNU6E5U4TDB/action/replication_record"}},"created_at":"2026-05-18T01:35:15.463548+00:00","updated_at":"2026-05-18T01:35:15.463548+00:00"}