{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:U5QMM3NDJ3NS5Q64IXBTO6DQWK","short_pith_number":"pith:U5QMM3ND","canonical_record":{"source":{"id":"1210.6568","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-10-24T15:18:59Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"da0ff6a04ddeba942dcf24a2c9f75e9302d0c199426b0804f2c8ccaad9e514c9","abstract_canon_sha256":"fbbc5030ab045b59803129cc4092627d022b94f8097ce39c6e6add50c4a4dc6d"},"schema_version":"1.0"},"canonical_sha256":"a760c66da34edb2ec3dc45c3377870b299a987231acd55da8f7f2e3ed7ec52cb","source":{"kind":"arxiv","id":"1210.6568","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.6568","created_at":"2026-05-18T03:42:24Z"},{"alias_kind":"arxiv_version","alias_value":"1210.6568v1","created_at":"2026-05-18T03:42:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.6568","created_at":"2026-05-18T03:42:24Z"},{"alias_kind":"pith_short_12","alias_value":"U5QMM3NDJ3NS","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"U5QMM3NDJ3NS5Q64","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"U5QMM3ND","created_at":"2026-05-18T12:27:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:U5QMM3NDJ3NS5Q64IXBTO6DQWK","target":"record","payload":{"canonical_record":{"source":{"id":"1210.6568","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-10-24T15:18:59Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"da0ff6a04ddeba942dcf24a2c9f75e9302d0c199426b0804f2c8ccaad9e514c9","abstract_canon_sha256":"fbbc5030ab045b59803129cc4092627d022b94f8097ce39c6e6add50c4a4dc6d"},"schema_version":"1.0"},"canonical_sha256":"a760c66da34edb2ec3dc45c3377870b299a987231acd55da8f7f2e3ed7ec52cb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:24.502626Z","signature_b64":"2wSzqJz7atiqhU7ryb7GD1qA2xvJ0N7ostrECBf5ffNOeel6gnG7Mi21zesgHPbDJwT6eXkl6Io8DfZU+/r2AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a760c66da34edb2ec3dc45c3377870b299a987231acd55da8f7f2e3ed7ec52cb","last_reissued_at":"2026-05-18T03:42:24.501957Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:24.501957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.6568","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:42:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MI5wGK5oxXj3qW0q8YSmNaqjovMEH3MFejZjOmZ3LfSj7ABBCEQZA2OF0FuUe7JfXcgRsBKW7W+1JFIFty0jCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:38:01.490396Z"},"content_sha256":"f57db9c55becf3557dee871932ad4708da45ec4f63e306095007b08430a676b1","schema_version":"1.0","event_id":"sha256:f57db9c55becf3557dee871932ad4708da45ec4f63e306095007b08430a676b1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:U5QMM3NDJ3NS5Q64IXBTO6DQWK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Equitable Colorings of Corona Multiproducts of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Hanna Furma\\~nczyk, Marek Kubale, Vahan V. Mkrtchyan","submitted_at":"2012-10-24T15:18:59Z","abstract_excerpt":"A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the equitable chromatic number of $G$ and denoted $\\chi_{=}(G)$. It is known that this problem is NP-hard in general case and remains so for corona graphs. In \"Equitable colorings of Cartesian products of graphs\" (2012) Lin and Chang studied equitable coloring of Cartesian products of graphs. In this paper we consider the same model of coloring in the case of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6568","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:42:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ccyRESFyE2SdcGTv47tx7tJPaN8lfwWigM3miVaV3qzdJDBmI1o5uB0WtIYYYGol2zRU10Jl/zqTNUGcHNIbCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:38:01.491052Z"},"content_sha256":"6636d41b62dafed9b313c68648ec0451882c82b80c36fcfa674cda37571fed0f","schema_version":"1.0","event_id":"sha256:6636d41b62dafed9b313c68648ec0451882c82b80c36fcfa674cda37571fed0f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U5QMM3NDJ3NS5Q64IXBTO6DQWK/bundle.json","state_url":"https://pith.science/pith/U5QMM3NDJ3NS5Q64IXBTO6DQWK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U5QMM3NDJ3NS5Q64IXBTO6DQWK/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T07:38:01Z","links":{"resolver":"https://pith.science/pith/U5QMM3NDJ3NS5Q64IXBTO6DQWK","bundle":"https://pith.science/pith/U5QMM3NDJ3NS5Q64IXBTO6DQWK/bundle.json","state":"https://pith.science/pith/U5QMM3NDJ3NS5Q64IXBTO6DQWK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U5QMM3NDJ3NS5Q64IXBTO6DQWK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:U5QMM3NDJ3NS5Q64IXBTO6DQWK","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":"fbbc5030ab045b59803129cc4092627d022b94f8097ce39c6e6add50c4a4dc6d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-10-24T15:18:59Z","title_canon_sha256":"da0ff6a04ddeba942dcf24a2c9f75e9302d0c199426b0804f2c8ccaad9e514c9"},"schema_version":"1.0","source":{"id":"1210.6568","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.6568","created_at":"2026-05-18T03:42:24Z"},{"alias_kind":"arxiv_version","alias_value":"1210.6568v1","created_at":"2026-05-18T03:42:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.6568","created_at":"2026-05-18T03:42:24Z"},{"alias_kind":"pith_short_12","alias_value":"U5QMM3NDJ3NS","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"U5QMM3NDJ3NS5Q64","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"U5QMM3ND","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:6636d41b62dafed9b313c68648ec0451882c82b80c36fcfa674cda37571fed0f","target":"graph","created_at":"2026-05-18T03:42:24Z","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 graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the equitable chromatic number of $G$ and denoted $\\chi_{=}(G)$. It is known that this problem is NP-hard in general case and remains so for corona graphs. In \"Equitable colorings of Cartesian products of graphs\" (2012) Lin and Chang studied equitable coloring of Cartesian products of graphs. In this paper we consider the same model of coloring in the case of","authors_text":"Hanna Furma\\~nczyk, Marek Kubale, Vahan V. Mkrtchyan","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-10-24T15:18:59Z","title":"Equitable Colorings of Corona Multiproducts of Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6568","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:f57db9c55becf3557dee871932ad4708da45ec4f63e306095007b08430a676b1","target":"record","created_at":"2026-05-18T03:42:24Z","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":"fbbc5030ab045b59803129cc4092627d022b94f8097ce39c6e6add50c4a4dc6d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-10-24T15:18:59Z","title_canon_sha256":"da0ff6a04ddeba942dcf24a2c9f75e9302d0c199426b0804f2c8ccaad9e514c9"},"schema_version":"1.0","source":{"id":"1210.6568","kind":"arxiv","version":1}},"canonical_sha256":"a760c66da34edb2ec3dc45c3377870b299a987231acd55da8f7f2e3ed7ec52cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a760c66da34edb2ec3dc45c3377870b299a987231acd55da8f7f2e3ed7ec52cb","first_computed_at":"2026-05-18T03:42:24.501957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:24.501957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2wSzqJz7atiqhU7ryb7GD1qA2xvJ0N7ostrECBf5ffNOeel6gnG7Mi21zesgHPbDJwT6eXkl6Io8DfZU+/r2AA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:24.502626Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.6568","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f57db9c55becf3557dee871932ad4708da45ec4f63e306095007b08430a676b1","sha256:6636d41b62dafed9b313c68648ec0451882c82b80c36fcfa674cda37571fed0f"],"state_sha256":"102055045fbf0168a728da020376790642938f59b155b605a848b15582331bcd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mSEOT/5UKv0IrQL3YspC/MQuZjFyeJO+EFY0xQZUiO7792/X6yiFh80sljNJi0g9yjniOmx7jdcp7rsckLMKBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T07:38:01.495289Z","bundle_sha256":"4bbfbe08d816bb28b308640f12276ce9617375f47fef5f04cbefb88988341160"}}