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That is, $\\phi$ is an m-clique hole if for all $i\\neq j$, $i,j=1,2,\\ldots,m$, $\\Phi_i \\cap \\Phi_{j} \\neq \\emptyset$ if and only if $(j-1)~\\mbox{mod}~m = (j+1)~\\mbox{mod}~m = i~\\mbox{mod}~m$. This paper derives a sufficient and necessary condition on m-colorability of m-clique holes, and proposes a coloring algorithm that colors m-clique holes with exactly m colors."},"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":"1508.06967","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-08-27T18:50:53Z","cross_cats_sorted":[],"title_canon_sha256":"4fceacde4742e7f6789798a379a643e741da46036471a1a726797560d42afa60","abstract_canon_sha256":"fa3e6153ccb29d30e3cb080b218b23e4c8ba8f945e96c964438f2379d593ce63"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:40.431506Z","signature_b64":"FQ0XxUpUBgAgSrU51kmdMx3BFsdh0Mnlxbmi0aGquS2PBqmUTQ57eg4bYYHDswQpkhH774fhTnifyHx3oyJtAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc5d15e02bcfe9959652fbbd194606c18233bccfb8cd74e7f459de0fadad0ae6","last_reissued_at":"2026-05-18T01:34:40.430711Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:40.430711Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Simple Algorithm for Coloring m-Clique Holes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Bechir Hamdaoui","submitted_at":"2015-08-27T18:50:53Z","abstract_excerpt":"An m-clique hole is a sequence $\\phi=(\\Phi_1,\\Phi_2,\\dots,\\Phi_m)$ of $m$ distinct cliques such that $|\\Phi_i| \\leq m$ for all $i=1,2,\\ldots,m$, and whose clique graph is a hole on $m$ vertices. That is, $\\phi$ is an m-clique hole if for all $i\\neq j$, $i,j=1,2,\\ldots,m$, $\\Phi_i \\cap \\Phi_{j} \\neq \\emptyset$ if and only if $(j-1)~\\mbox{mod}~m = (j+1)~\\mbox{mod}~m = i~\\mbox{mod}~m$. This paper derives a sufficient and necessary condition on m-colorability of m-clique holes, and proposes a coloring algorithm that colors m-clique holes with exactly m colors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06967","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":"1508.06967","created_at":"2026-05-18T01:34:40.430824+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.06967v1","created_at":"2026-05-18T01:34:40.430824+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06967","created_at":"2026-05-18T01:34:40.430824+00:00"},{"alias_kind":"pith_short_12","alias_value":"3RORLYBLZ7UZ","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"3RORLYBLZ7UZLFSS","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"3RORLYBL","created_at":"2026-05-18T12:29:02.477457+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/3RORLYBLZ7UZLFSS7O6RSRQGYG","json":"https://pith.science/pith/3RORLYBLZ7UZLFSS7O6RSRQGYG.json","graph_json":"https://pith.science/api/pith-number/3RORLYBLZ7UZLFSS7O6RSRQGYG/graph.json","events_json":"https://pith.science/api/pith-number/3RORLYBLZ7UZLFSS7O6RSRQGYG/events.json","paper":"https://pith.science/paper/3RORLYBL"},"agent_actions":{"view_html":"https://pith.science/pith/3RORLYBLZ7UZLFSS7O6RSRQGYG","download_json":"https://pith.science/pith/3RORLYBLZ7UZLFSS7O6RSRQGYG.json","view_paper":"https://pith.science/paper/3RORLYBL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.06967&json=true","fetch_graph":"https://pith.science/api/pith-number/3RORLYBLZ7UZLFSS7O6RSRQGYG/graph.json","fetch_events":"https://pith.science/api/pith-number/3RORLYBLZ7UZLFSS7O6RSRQGYG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3RORLYBLZ7UZLFSS7O6RSRQGYG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3RORLYBLZ7UZLFSS7O6RSRQGYG/action/storage_attestation","attest_author":"https://pith.science/pith/3RORLYBLZ7UZLFSS7O6RSRQGYG/action/author_attestation","sign_citation":"https://pith.science/pith/3RORLYBLZ7UZLFSS7O6RSRQGYG/action/citation_signature","submit_replication":"https://pith.science/pith/3RORLYBLZ7UZLFSS7O6RSRQGYG/action/replication_record"}},"created_at":"2026-05-18T01:34:40.430824+00:00","updated_at":"2026-05-18T01:34:40.430824+00:00"}