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We study the case when $G$ is the infinite hexagonal grid $H$. Cohen et.al. constructed two identifying codes for $H$ with density $3/7$ and proved that any identifying code for $H$ must have density at least $16/39\\approx0.410256$. Both their upper and lower bounds were best known until now. Here we prove a lower bound of $12/29\\approx0.4137"},"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":"1006.3779","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-18T19:26:36Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"c0b444374d46d2911bdd92b54421d40607a81e052e6d7adb38c6fbfed17f0872","abstract_canon_sha256":"fbe901c03d3f6d2c8b3371371700b499ab14dd51f30afa688b7537636eecbc31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:18.028421Z","signature_b64":"rdiOsr9si0GCFWpcU07tVOMhDcwV5NcO7IPe01HnkB5/M5wudOw7j/Ir7OVoXlVhK4r4oMw0REoguX5HQwi/DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7312ffa30fec992c46361842536859af4b2fbc0411b7763be7a975ce595e12a8","last_reissued_at":"2026-05-18T04:11:18.027817Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:18.027817Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A New Lower Bound on the Density of Vertex Identifying Codes for the Infinite Hexagonal Grid","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.CO","authors_text":"Daniel W. Cranston, Gexin Yu","submitted_at":"2010-06-18T19:26:36Z","abstract_excerpt":"Given a graph $G$, an identifying code $C \\subseteq V(G)$ is a vertex set such that for any two distinct vertices $v_1,v_2\\in V(G)$, the sets $N[v_1]\\cap C$ and $N[v_2]\\cap C$ are distinct and nonempty (here $N[v]$ denotes a vertex $v$ and its neighbors). We study the case when $G$ is the infinite hexagonal grid $H$. Cohen et.al. constructed two identifying codes for $H$ with density $3/7$ and proved that any identifying code for $H$ must have density at least $16/39\\approx0.410256$. Both their upper and lower bounds were best known until now. Here we prove a lower bound of $12/29\\approx0.4137"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.3779","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":"1006.3779","created_at":"2026-05-18T04:11:18.027912+00:00"},{"alias_kind":"arxiv_version","alias_value":"1006.3779v1","created_at":"2026-05-18T04:11:18.027912+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.3779","created_at":"2026-05-18T04:11:18.027912+00:00"},{"alias_kind":"pith_short_12","alias_value":"OMJP7IYP5SMS","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_16","alias_value":"OMJP7IYP5SMSYRRW","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_8","alias_value":"OMJP7IYP","created_at":"2026-05-18T12:26:12.377268+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2303.00557","citing_title":"Finding codes on infinite grids automatically","ref_index":4,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OMJP7IYP5SMSYRRWDBBFG2CZV5","json":"https://pith.science/pith/OMJP7IYP5SMSYRRWDBBFG2CZV5.json","graph_json":"https://pith.science/api/pith-number/OMJP7IYP5SMSYRRWDBBFG2CZV5/graph.json","events_json":"https://pith.science/api/pith-number/OMJP7IYP5SMSYRRWDBBFG2CZV5/events.json","paper":"https://pith.science/paper/OMJP7IYP"},"agent_actions":{"view_html":"https://pith.science/pith/OMJP7IYP5SMSYRRWDBBFG2CZV5","download_json":"https://pith.science/pith/OMJP7IYP5SMSYRRWDBBFG2CZV5.json","view_paper":"https://pith.science/paper/OMJP7IYP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1006.3779&json=true","fetch_graph":"https://pith.science/api/pith-number/OMJP7IYP5SMSYRRWDBBFG2CZV5/graph.json","fetch_events":"https://pith.science/api/pith-number/OMJP7IYP5SMSYRRWDBBFG2CZV5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OMJP7IYP5SMSYRRWDBBFG2CZV5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OMJP7IYP5SMSYRRWDBBFG2CZV5/action/storage_attestation","attest_author":"https://pith.science/pith/OMJP7IYP5SMSYRRWDBBFG2CZV5/action/author_attestation","sign_citation":"https://pith.science/pith/OMJP7IYP5SMSYRRWDBBFG2CZV5/action/citation_signature","submit_replication":"https://pith.science/pith/OMJP7IYP5SMSYRRWDBBFG2CZV5/action/replication_record"}},"created_at":"2026-05-18T04:11:18.027912+00:00","updated_at":"2026-05-18T04:11:18.027912+00:00"}