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A set $D \\subseteq V(G)$ is a vertex identifying code if for any two distinct vertices $v_1, v_2 \\in V(G)$, the vertex sets $N[v_1] \\cap D$ and $N[v_2] \\cap D$ are distinct and non-empty. We consider the minimum density of a vertex identifying code for the infinite hexagonal grid. In 2000, Cohen et al. constructed two codes with a density of $3/7 \\approx 0.428571$, and this remains the best known upper bound. Until now, the best known lower bound was $12/29 \\approx 0.413793$ and was pr"},"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":"1110.1097","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-05T20:28:53Z","cross_cats_sorted":[],"title_canon_sha256":"f0b41e4d3f135d000d5664a8b574bcc1bcbcd7ca0601d2c67bb92d7aacb8c783","abstract_canon_sha256":"ca7c7eedca25dbd59bf4ea045ad9a0c5bd3e177af792b57909b0a68d7b0fa463"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:39.302042Z","signature_b64":"gcGVGpz8q1P9CEWFFLVMtV4HNA0sJwpHZZn4XI3jS86ZuIEfqW+tUYX6nN9OyQQoyMTbrifptvflv7gwSyUJDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bfd30c1d51301ef0d4832abb67d316a56afa18a85325e572e9cf5719c6b3770f","last_reissued_at":"2026-05-18T04:11:39.301372Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:39.301372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New Bounds on the Minimum Density of a Vertex Identifying Code for the Infinite Hexagonal Grid","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ari Cukierman, Gexin Yu","submitted_at":"2011-10-05T20:28:53Z","abstract_excerpt":"For a graph, $G$, and a vertex $v \\in V(G)$, let $N[v]$ be the set of vertices adjacent to and including $v$. 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