{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:ICZPE5I6F2JQECKISFOF7KDKXV","short_pith_number":"pith:ICZPE5I6","canonical_record":{"source":{"id":"0911.4218","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-11-22T02:23:25Z","cross_cats_sorted":["cond-mat.stat-mech","math.MP"],"title_canon_sha256":"3368359528b00ae7e1db13dd92f807c77f16ccfe4376625571954573a502b385","abstract_canon_sha256":"7b6d4bb42d997cc428a1819915093e21aeff3c7dcbff228320d7862b771fcf6a"},"schema_version":"1.0"},"canonical_sha256":"40b2f2751e2e93020948915c5fa86abd6926b5729736e77e6c9720080f38e01f","source":{"kind":"arxiv","id":"0911.4218","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.4218","created_at":"2026-05-18T04:15:17Z"},{"alias_kind":"arxiv_version","alias_value":"0911.4218v1","created_at":"2026-05-18T04:15:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.4218","created_at":"2026-05-18T04:15:17Z"},{"alias_kind":"pith_short_12","alias_value":"ICZPE5I6F2JQ","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"ICZPE5I6F2JQECKI","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"ICZPE5I6","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:ICZPE5I6F2JQECKISFOF7KDKXV","target":"record","payload":{"canonical_record":{"source":{"id":"0911.4218","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-11-22T02:23:25Z","cross_cats_sorted":["cond-mat.stat-mech","math.MP"],"title_canon_sha256":"3368359528b00ae7e1db13dd92f807c77f16ccfe4376625571954573a502b385","abstract_canon_sha256":"7b6d4bb42d997cc428a1819915093e21aeff3c7dcbff228320d7862b771fcf6a"},"schema_version":"1.0"},"canonical_sha256":"40b2f2751e2e93020948915c5fa86abd6926b5729736e77e6c9720080f38e01f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:17.087513Z","signature_b64":"Zgm59C0NMP61SPeXlVznTqQZ9GnkjEMIfuVc2L4D72Zw6l8YVC2btVtM6WK6RaYeBHU17mqtnv0MacUxtdQZCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40b2f2751e2e93020948915c5fa86abd6926b5729736e77e6c9720080f38e01f","last_reissued_at":"2026-05-18T04:15:17.086978Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:17.086978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0911.4218","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":"2026-05-18T04:15:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tkQ66/hEA+4XFIZhGCtCkMgVjKdooIM9nNYXTxoE/EmPZ/hE3F5XMgSbd73rU3iSK4TtkX8mgeMFYuQNwN08BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T09:27:45.620420Z"},"content_sha256":"fa9b87a9b93742499c169d339247ab4f57484ca89e69e37557921022371b18fe","schema_version":"1.0","event_id":"sha256:fa9b87a9b93742499c169d339247ab4f57484ca89e69e37557921022371b18fe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:ICZPE5I6F2JQECKISFOF7KDKXV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted-Set Graph Colorings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math.MP"],"primary_cat":"math-ph","authors_text":"Robert Shrock, Yan Xu","submitted_at":"2009-11-22T02:23:25Z","abstract_excerpt":"We study a weighted-set graph coloring problem in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given subset of $s$ colors contained in the set of $q$ colors. We construct and analyze a weighted-set chromatic polynomial $Ph(G,q,s,w)$ associated with this coloring. General properties of this weighted-set chromatic polynomial are proved, and illustrative calculations are presented for various families of graphs. This study extends a previous one for the case $s=1$ and reveal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4218","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":"2026-05-18T04:15:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v/49Ufp4UoR95610+jM9W53D57pxg29IU1oPctsNGzs+8aQcli5CtjQHfb/Tpmi2gambfmjEgYS5PurPvSBmDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T09:27:45.620818Z"},"content_sha256":"baafea40b0d3ee2affc9320a271feadf339ae07e7467db6331109deb2fd5a5a3","schema_version":"1.0","event_id":"sha256:baafea40b0d3ee2affc9320a271feadf339ae07e7467db6331109deb2fd5a5a3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ICZPE5I6F2JQECKISFOF7KDKXV/bundle.json","state_url":"https://pith.science/pith/ICZPE5I6F2JQECKISFOF7KDKXV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ICZPE5I6F2JQECKISFOF7KDKXV/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":"2026-06-04T09:27:45Z","links":{"resolver":"https://pith.science/pith/ICZPE5I6F2JQECKISFOF7KDKXV","bundle":"https://pith.science/pith/ICZPE5I6F2JQECKISFOF7KDKXV/bundle.json","state":"https://pith.science/pith/ICZPE5I6F2JQECKISFOF7KDKXV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ICZPE5I6F2JQECKISFOF7KDKXV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:ICZPE5I6F2JQECKISFOF7KDKXV","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":"7b6d4bb42d997cc428a1819915093e21aeff3c7dcbff228320d7862b771fcf6a","cross_cats_sorted":["cond-mat.stat-mech","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-11-22T02:23:25Z","title_canon_sha256":"3368359528b00ae7e1db13dd92f807c77f16ccfe4376625571954573a502b385"},"schema_version":"1.0","source":{"id":"0911.4218","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.4218","created_at":"2026-05-18T04:15:17Z"},{"alias_kind":"arxiv_version","alias_value":"0911.4218v1","created_at":"2026-05-18T04:15:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.4218","created_at":"2026-05-18T04:15:17Z"},{"alias_kind":"pith_short_12","alias_value":"ICZPE5I6F2JQ","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"ICZPE5I6F2JQECKI","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"ICZPE5I6","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:baafea40b0d3ee2affc9320a271feadf339ae07e7467db6331109deb2fd5a5a3","target":"graph","created_at":"2026-05-18T04:15:17Z","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 study a weighted-set graph coloring problem in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given subset of $s$ colors contained in the set of $q$ colors. We construct and analyze a weighted-set chromatic polynomial $Ph(G,q,s,w)$ associated with this coloring. General properties of this weighted-set chromatic polynomial are proved, and illustrative calculations are presented for various families of graphs. This study extends a previous one for the case $s=1$ and reveal","authors_text":"Robert Shrock, Yan Xu","cross_cats":["cond-mat.stat-mech","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-11-22T02:23:25Z","title":"Weighted-Set Graph Colorings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4218","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:fa9b87a9b93742499c169d339247ab4f57484ca89e69e37557921022371b18fe","target":"record","created_at":"2026-05-18T04:15:17Z","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":"7b6d4bb42d997cc428a1819915093e21aeff3c7dcbff228320d7862b771fcf6a","cross_cats_sorted":["cond-mat.stat-mech","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-11-22T02:23:25Z","title_canon_sha256":"3368359528b00ae7e1db13dd92f807c77f16ccfe4376625571954573a502b385"},"schema_version":"1.0","source":{"id":"0911.4218","kind":"arxiv","version":1}},"canonical_sha256":"40b2f2751e2e93020948915c5fa86abd6926b5729736e77e6c9720080f38e01f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"40b2f2751e2e93020948915c5fa86abd6926b5729736e77e6c9720080f38e01f","first_computed_at":"2026-05-18T04:15:17.086978Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:15:17.086978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Zgm59C0NMP61SPeXlVznTqQZ9GnkjEMIfuVc2L4D72Zw6l8YVC2btVtM6WK6RaYeBHU17mqtnv0MacUxtdQZCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:15:17.087513Z","signed_message":"canonical_sha256_bytes"},"source_id":"0911.4218","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa9b87a9b93742499c169d339247ab4f57484ca89e69e37557921022371b18fe","sha256:baafea40b0d3ee2affc9320a271feadf339ae07e7467db6331109deb2fd5a5a3"],"state_sha256":"cbbcc623e8484027ba73a6076481dc4ae006c7039b20a8e1d7f2e43ebc5b1651"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZWKLt0Yeu7b0TmTuhG356T8i/qXGOobPVs3Q0gNBVhJisiBZz6rr1OLnom4/eYeapsnAGHDFxHUG+je3U9l4AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T09:27:45.622789Z","bundle_sha256":"db0e49e47cb8e427285954eff61e43bfd1f93ac7ef6e64e4b9f6f94b4d072d4c"}}