{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:VWPVJRN5PEIO42F2F2Z6QXN5AA","short_pith_number":"pith:VWPVJRN5","schema_version":"1.0","canonical_sha256":"ad9f54c5bd7910ee68ba2eb3e85dbd002bd111e1596abefd851619f90b4c678d","source":{"kind":"arxiv","id":"1104.4411","version":1},"attestation_state":"computed","paper":{"title":"On Coloring Properties of Graph Powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ali Taherkhani, Hossein Hajiabolhassan","submitted_at":"2011-04-22T08:42:31Z","abstract_excerpt":"This paper studies some coloring properties of graph powers. We show that $\\chi_c(G^{^{\\frac{2r+1}{2s+1}}})=\\frac{(2s+1)\\chi_c(G)}{(s-r)\\chi_c(G)+2r+1}$ provided that $\\chi_c(G^{^{\\frac{2r+1}{2s+1}}})< 4$. As a consequence, one can see that if ${2r+1 \\over 2s+1} \\leq {\\chi_c(G) \\over 3(\\chi_c(G)-2)}$, then $\\chi_c(G^{^{\\frac{2r+1}{2s+1}}})=\\frac{(2s+1)\\chi_c(G)}{(s-r)\\chi_c(G)+2r+1}$. In particular, $\\chi_c(K_{3n+1}^{^{1\\over3}})={9n+3\\over 3n+2}$ and $K_{3n+1}^{^{1\\over3}}$ has no subgraph with circular chromatic number equal to ${6n+1\\over 2n+1}$. This provides a negative answer to a questio"},"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":"1104.4411","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-22T08:42:31Z","cross_cats_sorted":[],"title_canon_sha256":"992ff8af59ec9615fd3af23d02932d483d41cd233db786da758ee2c006031a05","abstract_canon_sha256":"73086dcc9a0ba0f2cda6fce65f85b739520e88568f9e4be3c1cf941c4e7f2d95"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:34.657803Z","signature_b64":"87yihdDTYg4EqFVwR0jJjcpEnP0jHRrt/iMmraMd/ciEickA0foIPp+Lpr61V5A1hccUXjd802VVWQR1h7+cCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad9f54c5bd7910ee68ba2eb3e85dbd002bd111e1596abefd851619f90b4c678d","last_reissued_at":"2026-05-18T04:23:34.657356Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:34.657356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Coloring Properties of Graph Powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ali Taherkhani, Hossein Hajiabolhassan","submitted_at":"2011-04-22T08:42:31Z","abstract_excerpt":"This paper studies some coloring properties of graph powers. We show that $\\chi_c(G^{^{\\frac{2r+1}{2s+1}}})=\\frac{(2s+1)\\chi_c(G)}{(s-r)\\chi_c(G)+2r+1}$ provided that $\\chi_c(G^{^{\\frac{2r+1}{2s+1}}})< 4$. As a consequence, one can see that if ${2r+1 \\over 2s+1} \\leq {\\chi_c(G) \\over 3(\\chi_c(G)-2)}$, then $\\chi_c(G^{^{\\frac{2r+1}{2s+1}}})=\\frac{(2s+1)\\chi_c(G)}{(s-r)\\chi_c(G)+2r+1}$. In particular, $\\chi_c(K_{3n+1}^{^{1\\over3}})={9n+3\\over 3n+2}$ and $K_{3n+1}^{^{1\\over3}}$ has no subgraph with circular chromatic number equal to ${6n+1\\over 2n+1}$. This provides a negative answer to a questio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4411","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":"1104.4411","created_at":"2026-05-18T04:23:34.657413+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.4411v1","created_at":"2026-05-18T04:23:34.657413+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.4411","created_at":"2026-05-18T04:23:34.657413+00:00"},{"alias_kind":"pith_short_12","alias_value":"VWPVJRN5PEIO","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_16","alias_value":"VWPVJRN5PEIO42F2","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_8","alias_value":"VWPVJRN5","created_at":"2026-05-18T12:26:44.992195+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/VWPVJRN5PEIO42F2F2Z6QXN5AA","json":"https://pith.science/pith/VWPVJRN5PEIO42F2F2Z6QXN5AA.json","graph_json":"https://pith.science/api/pith-number/VWPVJRN5PEIO42F2F2Z6QXN5AA/graph.json","events_json":"https://pith.science/api/pith-number/VWPVJRN5PEIO42F2F2Z6QXN5AA/events.json","paper":"https://pith.science/paper/VWPVJRN5"},"agent_actions":{"view_html":"https://pith.science/pith/VWPVJRN5PEIO42F2F2Z6QXN5AA","download_json":"https://pith.science/pith/VWPVJRN5PEIO42F2F2Z6QXN5AA.json","view_paper":"https://pith.science/paper/VWPVJRN5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.4411&json=true","fetch_graph":"https://pith.science/api/pith-number/VWPVJRN5PEIO42F2F2Z6QXN5AA/graph.json","fetch_events":"https://pith.science/api/pith-number/VWPVJRN5PEIO42F2F2Z6QXN5AA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VWPVJRN5PEIO42F2F2Z6QXN5AA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VWPVJRN5PEIO42F2F2Z6QXN5AA/action/storage_attestation","attest_author":"https://pith.science/pith/VWPVJRN5PEIO42F2F2Z6QXN5AA/action/author_attestation","sign_citation":"https://pith.science/pith/VWPVJRN5PEIO42F2F2Z6QXN5AA/action/citation_signature","submit_replication":"https://pith.science/pith/VWPVJRN5PEIO42F2F2Z6QXN5AA/action/replication_record"}},"created_at":"2026-05-18T04:23:34.657413+00:00","updated_at":"2026-05-18T04:23:34.657413+00:00"}