{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:I3LJ7UEQNPXADXCRDJX3NYQ3DY","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":"8f29e3eca5fd753bf3663b27e27c0c453201c7eaf16db43dcb1a72e6ec42edd4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-08-29T21:22:34Z","title_canon_sha256":"4d0919d29d4e716bc7b9bfd63c68b0ce45627ddfbb984c4bd0d4e70e1d899f07"},"schema_version":"1.0","source":{"id":"1108.5757","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.5757","created_at":"2026-05-18T00:59:42Z"},{"alias_kind":"arxiv_version","alias_value":"1108.5757v1","created_at":"2026-05-18T00:59:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5757","created_at":"2026-05-18T00:59:42Z"},{"alias_kind":"pith_short_12","alias_value":"I3LJ7UEQNPXA","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"I3LJ7UEQNPXADXCR","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"I3LJ7UEQ","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:06350da7434258ebc25a728dc9f9868d23c83575e95f54121649b5bdb8df43e2","target":"graph","created_at":"2026-05-18T00:59:42Z","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":"A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, ..., x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by \\chi_k(G). We determine the exact value of this parameter when G is a web or an antiweb. Our results generalize the known corresponding results for odd cycles and imply necessary and sufficient conditions under which \\chi_k(G) attains its lower and upper bounds based on the clique, the fractiona","authors_text":"Manoel Camp\\^elo, Marcio C. Santos, Phablo F. S. Moura, Ricardo C. Corr\\^ea","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-08-29T21:22:34Z","title":"Optimal k-fold colorings of webs and antiwebs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5757","kind":"arxiv","version":1},"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":"sha256:307786f44613da61e3bbb82967651a75085e811b0b988cc39fcbaf59bb937f8a","target":"record","created_at":"2026-05-18T00:59:42Z","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":"8f29e3eca5fd753bf3663b27e27c0c453201c7eaf16db43dcb1a72e6ec42edd4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-08-29T21:22:34Z","title_canon_sha256":"4d0919d29d4e716bc7b9bfd63c68b0ce45627ddfbb984c4bd0d4e70e1d899f07"},"schema_version":"1.0","source":{"id":"1108.5757","kind":"arxiv","version":1}},"canonical_sha256":"46d69fd0906bee01dc511a6fb6e21b1e04ab3988b2b362554d8a524a6160375b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"46d69fd0906bee01dc511a6fb6e21b1e04ab3988b2b362554d8a524a6160375b","first_computed_at":"2026-05-18T00:59:42.582783Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:59:42.582783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wYgu7P7t2f/JERcmJ5FkzuPg14MQZ3KNV+0obq8lQO7n6zHOUIHJlcdLhCAB5lw2+3f7ZOin3WDpgXYSErsABQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:59:42.583309Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.5757","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:307786f44613da61e3bbb82967651a75085e811b0b988cc39fcbaf59bb937f8a","sha256:06350da7434258ebc25a728dc9f9868d23c83575e95f54121649b5bdb8df43e2"],"state_sha256":"14271978c297c2e32d884cb266fbaa6548fcfd1b1e837ea8c6488bd902a297bc"}