{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:O5X25HAEDLIWRAUL6ZCUIPHYOH","short_pith_number":"pith:O5X25HAE","canonical_record":{"source":{"id":"1505.06234","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-22T21:55:20Z","cross_cats_sorted":[],"title_canon_sha256":"742f98ca5f472c3773bacd935c13cb13ef4e0052c309c123dc56c0321c7c3a0c","abstract_canon_sha256":"6287965df7856e1802df804467f16442115bd3005e6194c54fe80292817445d1"},"schema_version":"1.0"},"canonical_sha256":"776fae9c041ad168828bf645443cf871dcf6d07000baf601d859f61867078f57","source":{"kind":"arxiv","id":"1505.06234","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.06234","created_at":"2026-05-18T00:54:15Z"},{"alias_kind":"arxiv_version","alias_value":"1505.06234v3","created_at":"2026-05-18T00:54:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.06234","created_at":"2026-05-18T00:54:15Z"},{"alias_kind":"pith_short_12","alias_value":"O5X25HAEDLIW","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"O5X25HAEDLIWRAUL","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"O5X25HAE","created_at":"2026-05-18T12:29:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:O5X25HAEDLIWRAUL6ZCUIPHYOH","target":"record","payload":{"canonical_record":{"source":{"id":"1505.06234","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-22T21:55:20Z","cross_cats_sorted":[],"title_canon_sha256":"742f98ca5f472c3773bacd935c13cb13ef4e0052c309c123dc56c0321c7c3a0c","abstract_canon_sha256":"6287965df7856e1802df804467f16442115bd3005e6194c54fe80292817445d1"},"schema_version":"1.0"},"canonical_sha256":"776fae9c041ad168828bf645443cf871dcf6d07000baf601d859f61867078f57","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:15.909473Z","signature_b64":"s6gkzC2jkBfqH/AqtFIWPL8qfg1s1Bz+LT8V9qNvCkZhgX9d50UsVQxNXrk5M/zLl2K9OuB3G15w/Y72hWSSCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"776fae9c041ad168828bf645443cf871dcf6d07000baf601d859f61867078f57","last_reissued_at":"2026-05-18T00:54:15.909016Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:15.909016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.06234","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:54:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hb/YFYO/25cE0qt6PMqH/VWd82DJ6MrKTY2iIXNg9QVm/DQGq5xUi/+nGtm9Vu93BdBxsVKRQHgZ4GcwIEHNAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T01:28:33.622036Z"},"content_sha256":"e19d85d5ad4d01efedcdc9e264ef1db941463ca27e3e61f375212f919ec19df7","schema_version":"1.0","event_id":"sha256:e19d85d5ad4d01efedcdc9e264ef1db941463ca27e3e61f375212f919ec19df7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:O5X25HAEDLIWRAUL6ZCUIPHYOH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tree-chromatic number is not equal to path-chromatic number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ringi Kim, Tony Huynh","submitted_at":"2015-05-22T21:55:20Z","abstract_excerpt":"For a graph $G$ and a tree-decomposition $(T, \\mathcal{B})$ of $G$, the chromatic number of $(T, \\mathcal{B})$ is the maximum of $\\chi(G[B])$, taken over all bags $B \\in \\mathcal{B}$. The tree-chromatic number of $G$ is the minimum chromatic number of all tree-decompositions $(T, \\mathcal{B})$ of $G$. The path-chromatic number of $G$ is defined analogously. In this paper, we introduce an operation that always increases the path-chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path-chromatic number and tree-chromatic number are "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06234","kind":"arxiv","version":3},"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":"2026-05-18T00:54:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lGQMC3zsnLrTo/gNNZEdHd0ReyAM1MSmpcxShjqT6xGNv7xIu/IriYxjCOWGXS1iSrILbyH8sOvlIb/vX/apBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T01:28:33.622728Z"},"content_sha256":"8ab9011558668b835b014caca0d008b1d68266875cbfe8eb695c665f6c443e29","schema_version":"1.0","event_id":"sha256:8ab9011558668b835b014caca0d008b1d68266875cbfe8eb695c665f6c443e29"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O5X25HAEDLIWRAUL6ZCUIPHYOH/bundle.json","state_url":"https://pith.science/pith/O5X25HAEDLIWRAUL6ZCUIPHYOH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O5X25HAEDLIWRAUL6ZCUIPHYOH/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":"2026-06-11T01:28:33Z","links":{"resolver":"https://pith.science/pith/O5X25HAEDLIWRAUL6ZCUIPHYOH","bundle":"https://pith.science/pith/O5X25HAEDLIWRAUL6ZCUIPHYOH/bundle.json","state":"https://pith.science/pith/O5X25HAEDLIWRAUL6ZCUIPHYOH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O5X25HAEDLIWRAUL6ZCUIPHYOH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:O5X25HAEDLIWRAUL6ZCUIPHYOH","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":"6287965df7856e1802df804467f16442115bd3005e6194c54fe80292817445d1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-22T21:55:20Z","title_canon_sha256":"742f98ca5f472c3773bacd935c13cb13ef4e0052c309c123dc56c0321c7c3a0c"},"schema_version":"1.0","source":{"id":"1505.06234","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.06234","created_at":"2026-05-18T00:54:15Z"},{"alias_kind":"arxiv_version","alias_value":"1505.06234v3","created_at":"2026-05-18T00:54:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.06234","created_at":"2026-05-18T00:54:15Z"},{"alias_kind":"pith_short_12","alias_value":"O5X25HAEDLIW","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"O5X25HAEDLIWRAUL","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"O5X25HAE","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:8ab9011558668b835b014caca0d008b1d68266875cbfe8eb695c665f6c443e29","target":"graph","created_at":"2026-05-18T00:54:15Z","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":"For a graph $G$ and a tree-decomposition $(T, \\mathcal{B})$ of $G$, the chromatic number of $(T, \\mathcal{B})$ is the maximum of $\\chi(G[B])$, taken over all bags $B \\in \\mathcal{B}$. The tree-chromatic number of $G$ is the minimum chromatic number of all tree-decompositions $(T, \\mathcal{B})$ of $G$. The path-chromatic number of $G$ is defined analogously. In this paper, we introduce an operation that always increases the path-chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path-chromatic number and tree-chromatic number are ","authors_text":"Ringi Kim, Tony Huynh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-22T21:55:20Z","title":"Tree-chromatic number is not equal to path-chromatic number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06234","kind":"arxiv","version":3},"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:e19d85d5ad4d01efedcdc9e264ef1db941463ca27e3e61f375212f919ec19df7","target":"record","created_at":"2026-05-18T00:54:15Z","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":"6287965df7856e1802df804467f16442115bd3005e6194c54fe80292817445d1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-22T21:55:20Z","title_canon_sha256":"742f98ca5f472c3773bacd935c13cb13ef4e0052c309c123dc56c0321c7c3a0c"},"schema_version":"1.0","source":{"id":"1505.06234","kind":"arxiv","version":3}},"canonical_sha256":"776fae9c041ad168828bf645443cf871dcf6d07000baf601d859f61867078f57","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"776fae9c041ad168828bf645443cf871dcf6d07000baf601d859f61867078f57","first_computed_at":"2026-05-18T00:54:15.909016Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:54:15.909016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s6gkzC2jkBfqH/AqtFIWPL8qfg1s1Bz+LT8V9qNvCkZhgX9d50UsVQxNXrk5M/zLl2K9OuB3G15w/Y72hWSSCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:54:15.909473Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.06234","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e19d85d5ad4d01efedcdc9e264ef1db941463ca27e3e61f375212f919ec19df7","sha256:8ab9011558668b835b014caca0d008b1d68266875cbfe8eb695c665f6c443e29"],"state_sha256":"db0d5d6a5c93b9cd332c8a16da62483a4156af65ab7f32bca956da299c24aedc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mmygpnk+zSP0XuFfiinpvLvKnCIa8QxE+rJBi2zcOwAkgbJ6i53vSJIhxfI9Y3VBgDGTSH5H5ySyMrCgErjvCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T01:28:33.626949Z","bundle_sha256":"685ec508c5458a7ae325668e6af2037686b35f527e54674aaa28836c9409aeef"}}