{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:DS7VIYDHMXQ5PCPREGPYPVN5E2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"275ab493af04485660554afb1dd3603b01a2aa97c1a7b2781c40d6bfacb14fb9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2018-05-14T04:14:49Z","title_canon_sha256":"31bdc97883df7bc387e6d3aac16b82bedb3b6acf8b4318f23b93363265edeff5"},"schema_version":"1.0","source":{"id":"1805.05007","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.05007","created_at":"2026-05-17T23:57:02Z"},{"alias_kind":"arxiv_version","alias_value":"1805.05007v2","created_at":"2026-05-17T23:57:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.05007","created_at":"2026-05-17T23:57:02Z"},{"alias_kind":"pith_short_12","alias_value":"DS7VIYDHMXQ5","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DS7VIYDHMXQ5PCPR","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DS7VIYDH","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:fb0c8f96b2eb051581cba04b5744a863d22ca2d8804cc7b65a03893c18281b49","target":"graph","created_at":"2026-05-17T23:57:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We elucidate the structure of $(P_6,C_4)$-free graphs by showing that every such graph either has a clique cutset, or a universal vertex, or belongs to several special classes of graphs. Using this result, we show that for any $(P_6,C_4)$-free graph $G$, $\\lceil\\frac{5\\omega(G)}{4}\\rceil$ and $\\lceil\\frac{\\Delta(G) + \\omega(G) +1}{2}\\rceil$ are tight upper bounds for the chromatic number of $G$. Moreover, our structural results imply that every ($P_6$,$C_4$)-free graph with no clique cutset has bounded clique-width, and thus the existence of a polynomial-time algorithm that computes the chroma","authors_text":"Frederic Maffray, T. Karthick","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2018-05-14T04:14:49Z","title":"Square-free graphs with no six-vertex induced path"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.05007","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9ba22258f37c45cabfd3473590acb776acda8236fd7094bba396853f1f61eaa4","target":"record","created_at":"2026-05-17T23:57:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"275ab493af04485660554afb1dd3603b01a2aa97c1a7b2781c40d6bfacb14fb9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2018-05-14T04:14:49Z","title_canon_sha256":"31bdc97883df7bc387e6d3aac16b82bedb3b6acf8b4318f23b93363265edeff5"},"schema_version":"1.0","source":{"id":"1805.05007","kind":"arxiv","version":2}},"canonical_sha256":"1cbf54606765e1d789f1219f87d5bd26a6c35abf772a9ccece243881457e6165","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1cbf54606765e1d789f1219f87d5bd26a6c35abf772a9ccece243881457e6165","first_computed_at":"2026-05-17T23:57:02.957618Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:02.957618Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rdKFLuwePn6X4o9j+wIjmsAu9PE+DibTiZ/aSeWr7j8bdr1d5O4toypYP86tvA6AhGx3UrC2BepSi8JQe52OBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:02.958181Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.05007","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9ba22258f37c45cabfd3473590acb776acda8236fd7094bba396853f1f61eaa4","sha256:fb0c8f96b2eb051581cba04b5744a863d22ca2d8804cc7b65a03893c18281b49"],"state_sha256":"fe6581c91b3eb76a0a58900b93293b3c7e4f560bfc4b4140dffbb3d7610ad6e4"}