{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:CB2L63E7M44OTQ7MEF2HYDEE6Q","short_pith_number":"pith:CB2L63E7","canonical_record":{"source":{"id":"1107.2161","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-11T22:35:14Z","cross_cats_sorted":[],"title_canon_sha256":"5b8ae8b1c1c940b8f70beb55d3736994c27cc3eb4f6ed77f5d3c4ea42d3ae63e","abstract_canon_sha256":"f91779a0a7a5810cc7b98b6ed23beb6fcac32f664d9a1008349e52a747be9c6e"},"schema_version":"1.0"},"canonical_sha256":"1074bf6c9f6738e9c3ec21747c0c84f401fb0add8a6aaa5ea1cde9a3d0e2d47e","source":{"kind":"arxiv","id":"1107.2161","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2161","created_at":"2026-05-18T04:18:32Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2161v1","created_at":"2026-05-18T04:18:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2161","created_at":"2026-05-18T04:18:32Z"},{"alias_kind":"pith_short_12","alias_value":"CB2L63E7M44O","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"CB2L63E7M44OTQ7M","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"CB2L63E7","created_at":"2026-05-18T12:26:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:CB2L63E7M44OTQ7MEF2HYDEE6Q","target":"record","payload":{"canonical_record":{"source":{"id":"1107.2161","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-11T22:35:14Z","cross_cats_sorted":[],"title_canon_sha256":"5b8ae8b1c1c940b8f70beb55d3736994c27cc3eb4f6ed77f5d3c4ea42d3ae63e","abstract_canon_sha256":"f91779a0a7a5810cc7b98b6ed23beb6fcac32f664d9a1008349e52a747be9c6e"},"schema_version":"1.0"},"canonical_sha256":"1074bf6c9f6738e9c3ec21747c0c84f401fb0add8a6aaa5ea1cde9a3d0e2d47e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:32.779670Z","signature_b64":"xtFfyuk8cDpSttrMT0Mph7DbIkZw7mA7v/ZQiAipWGZ1LVPhXSU48ifcHBqV/0FqfM8w9MlYXh26Ya22ZReZCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1074bf6c9f6738e9c3ec21747c0c84f401fb0add8a6aaa5ea1cde9a3d0e2d47e","last_reissued_at":"2026-05-18T04:18:32.779185Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:32.779185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.2161","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":"2026-05-18T04:18:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yo6GDouBJV8/NIONLt+yFoVOwb7mb24g+rYLTxCfCkqIyjSf9afmOZ0AFJA0Q46sPWktmoplS+khlEgX3CwIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:58:53.053149Z"},"content_sha256":"d8248bc5a409ffbd395cd6b53e113031680490f41d69b38daadbdb18451f8d4d","schema_version":"1.0","event_id":"sha256:d8248bc5a409ffbd395cd6b53e113031680490f41d69b38daadbdb18451f8d4d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:CB2L63E7M44OTQ7MEF2HYDEE6Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Classes of graphs with small rank decompositions are chi-bounded","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel Kral, Zdenek Dvorak","submitted_at":"2011-07-11T22:35:14Z","abstract_excerpt":"A class of graphs G is chi-bounded if the chromatic number of graphs in G is bounded by a function of the clique number. We show that if a class G is chi-bounded,then every class of graphs admitting a decomposition along cuts of small rank to graphs from G is chi-bounded. As a corollary, we obtain that every class of graphs with bounded rank-width (or equivalently, clique-width) is chi-bounded."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2161","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":"2026-05-18T04:18:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DYiNz6BCeCrOnTeYmWgpc1yq6XH9YReoz+hqyVtKquHdvoe3Bd28jcLudoRvYD40e3BworbpRsLNHk0hi8uJBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:58:53.053743Z"},"content_sha256":"ad16ca2b59172bfc0fcf97c215c764c20df77783ed44f5d4627c56ec92cdd8bf","schema_version":"1.0","event_id":"sha256:ad16ca2b59172bfc0fcf97c215c764c20df77783ed44f5d4627c56ec92cdd8bf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CB2L63E7M44OTQ7MEF2HYDEE6Q/bundle.json","state_url":"https://pith.science/pith/CB2L63E7M44OTQ7MEF2HYDEE6Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CB2L63E7M44OTQ7MEF2HYDEE6Q/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":"2026-05-25T16:58:53Z","links":{"resolver":"https://pith.science/pith/CB2L63E7M44OTQ7MEF2HYDEE6Q","bundle":"https://pith.science/pith/CB2L63E7M44OTQ7MEF2HYDEE6Q/bundle.json","state":"https://pith.science/pith/CB2L63E7M44OTQ7MEF2HYDEE6Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CB2L63E7M44OTQ7MEF2HYDEE6Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:CB2L63E7M44OTQ7MEF2HYDEE6Q","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":"f91779a0a7a5810cc7b98b6ed23beb6fcac32f664d9a1008349e52a747be9c6e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-11T22:35:14Z","title_canon_sha256":"5b8ae8b1c1c940b8f70beb55d3736994c27cc3eb4f6ed77f5d3c4ea42d3ae63e"},"schema_version":"1.0","source":{"id":"1107.2161","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2161","created_at":"2026-05-18T04:18:32Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2161v1","created_at":"2026-05-18T04:18:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2161","created_at":"2026-05-18T04:18:32Z"},{"alias_kind":"pith_short_12","alias_value":"CB2L63E7M44O","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"CB2L63E7M44OTQ7M","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"CB2L63E7","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:ad16ca2b59172bfc0fcf97c215c764c20df77783ed44f5d4627c56ec92cdd8bf","target":"graph","created_at":"2026-05-18T04:18:32Z","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 class of graphs G is chi-bounded if the chromatic number of graphs in G is bounded by a function of the clique number. We show that if a class G is chi-bounded,then every class of graphs admitting a decomposition along cuts of small rank to graphs from G is chi-bounded. As a corollary, we obtain that every class of graphs with bounded rank-width (or equivalently, clique-width) is chi-bounded.","authors_text":"Daniel Kral, Zdenek Dvorak","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-11T22:35:14Z","title":"Classes of graphs with small rank decompositions are chi-bounded"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2161","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:d8248bc5a409ffbd395cd6b53e113031680490f41d69b38daadbdb18451f8d4d","target":"record","created_at":"2026-05-18T04:18:32Z","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":"f91779a0a7a5810cc7b98b6ed23beb6fcac32f664d9a1008349e52a747be9c6e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-11T22:35:14Z","title_canon_sha256":"5b8ae8b1c1c940b8f70beb55d3736994c27cc3eb4f6ed77f5d3c4ea42d3ae63e"},"schema_version":"1.0","source":{"id":"1107.2161","kind":"arxiv","version":1}},"canonical_sha256":"1074bf6c9f6738e9c3ec21747c0c84f401fb0add8a6aaa5ea1cde9a3d0e2d47e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1074bf6c9f6738e9c3ec21747c0c84f401fb0add8a6aaa5ea1cde9a3d0e2d47e","first_computed_at":"2026-05-18T04:18:32.779185Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:32.779185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xtFfyuk8cDpSttrMT0Mph7DbIkZw7mA7v/ZQiAipWGZ1LVPhXSU48ifcHBqV/0FqfM8w9MlYXh26Ya22ZReZCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:32.779670Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.2161","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d8248bc5a409ffbd395cd6b53e113031680490f41d69b38daadbdb18451f8d4d","sha256:ad16ca2b59172bfc0fcf97c215c764c20df77783ed44f5d4627c56ec92cdd8bf"],"state_sha256":"75684fed7c2f2eaab158a2b0685378bfa46c5392203b2b52067b272c628483ba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k12GZx8P8MZCysVZtgBFH2SFNikOpJgxvV5M3G537T8S7otNN8+pUXQD+dG0a7xc1nSAo2DM8K+FonOecXTlBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T16:58:53.057443Z","bundle_sha256":"aea5cbc9cacc026ac1dc8c9b444bbad925e80fda93d8e9d00c93fbcaa47859ec"}}