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The incidence chromatic number of $G$ is the smallest $k$ for which there exists a mapping from the set of incidences of $G$ to a set of $k$ colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid $T_{m,n}=C_m\\Box C_n$ equals 5 when $m,n \\equiv 0 \\pmod 5$ and 6 otherwise."},"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":"0907.3801","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2009-07-22T09:08:11Z","cross_cats_sorted":[],"title_canon_sha256":"97e305e9f10e6426a97c04fbd0cc6cf048ed2daa7a93680fe551d893bc173516","abstract_canon_sha256":"99c7fb76a7388b7770bf6c61a8ea52ab10e9f3b7a91d39e09d72baf2790e4036"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:19.953852Z","signature_b64":"ubysum6unU5us/Mbe1tBepmEx4YXTqBTjxfbQzCezUNdFiLR/F3NICFmWmez8Rjq9En+LmoOxp3/OjYYMymzCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"83e1e96b94f6598ade4c5df9d20d76bfeaae3b645af321f1f430796a6979c3fe","last_reissued_at":"2026-05-18T03:28:19.953401Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:19.953401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Incidence Chromatic Number of Toroidal Grids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Eric Sopena (LaBRI), Jiaojiao Wu (LaBRI)","submitted_at":"2009-07-22T09:08:11Z","abstract_excerpt":"An incidence in a graph $G$ is a pair $(v,e)$ with $v \\in V(G)$ and $e \\in E(G)$, such that $v$ and $e$ are incident. 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In this paper, we prove that the incidence chromatic number of the toroidal grid $T_{m,n}=C_m\\Box C_n$ equals 5 when $m,n \\equiv 0 \\pmod 5$ and 6 otherwise."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.3801","kind":"arxiv","version":3},"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":"0907.3801","created_at":"2026-05-18T03:28:19.953466+00:00"},{"alias_kind":"arxiv_version","alias_value":"0907.3801v3","created_at":"2026-05-18T03:28:19.953466+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.3801","created_at":"2026-05-18T03:28:19.953466+00:00"},{"alias_kind":"pith_short_12","alias_value":"QPQ6S24U6ZMY","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_16","alias_value":"QPQ6S24U6ZMYVXSM","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_8","alias_value":"QPQ6S24U","created_at":"2026-05-18T12:26:01.383474+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/QPQ6S24U6ZMYVXSMLX45EDLWX7","json":"https://pith.science/pith/QPQ6S24U6ZMYVXSMLX45EDLWX7.json","graph_json":"https://pith.science/api/pith-number/QPQ6S24U6ZMYVXSMLX45EDLWX7/graph.json","events_json":"https://pith.science/api/pith-number/QPQ6S24U6ZMYVXSMLX45EDLWX7/events.json","paper":"https://pith.science/paper/QPQ6S24U"},"agent_actions":{"view_html":"https://pith.science/pith/QPQ6S24U6ZMYVXSMLX45EDLWX7","download_json":"https://pith.science/pith/QPQ6S24U6ZMYVXSMLX45EDLWX7.json","view_paper":"https://pith.science/paper/QPQ6S24U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0907.3801&json=true","fetch_graph":"https://pith.science/api/pith-number/QPQ6S24U6ZMYVXSMLX45EDLWX7/graph.json","fetch_events":"https://pith.science/api/pith-number/QPQ6S24U6ZMYVXSMLX45EDLWX7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QPQ6S24U6ZMYVXSMLX45EDLWX7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QPQ6S24U6ZMYVXSMLX45EDLWX7/action/storage_attestation","attest_author":"https://pith.science/pith/QPQ6S24U6ZMYVXSMLX45EDLWX7/action/author_attestation","sign_citation":"https://pith.science/pith/QPQ6S24U6ZMYVXSMLX45EDLWX7/action/citation_signature","submit_replication":"https://pith.science/pith/QPQ6S24U6ZMYVXSMLX45EDLWX7/action/replication_record"}},"created_at":"2026-05-18T03:28:19.953466+00:00","updated_at":"2026-05-18T03:28:19.953466+00:00"}