{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:ECPHSAK344C3KIDEQFVPC2BVUZ","short_pith_number":"pith:ECPHSAK3","canonical_record":{"source":{"id":"2303.05505","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-09T18:56:26Z","cross_cats_sorted":[],"title_canon_sha256":"2f72e865ee266799e729e6183e480fc9910bbffe3c42362ced34ac552f50e431","abstract_canon_sha256":"2e0b50e7ca89a4004c0ca007e7b7f4956c56ecba1a16a67420dc738368e3092d"},"schema_version":"1.0"},"canonical_sha256":"209e79015be705b52064816af16835a660b8982dd63a5fdb073b96358ff520b6","source":{"kind":"arxiv","id":"2303.05505","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.05505","created_at":"2026-07-05T06:14:41Z"},{"alias_kind":"arxiv_version","alias_value":"2303.05505v2","created_at":"2026-07-05T06:14:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.05505","created_at":"2026-07-05T06:14:41Z"},{"alias_kind":"pith_short_12","alias_value":"ECPHSAK344C3","created_at":"2026-07-05T06:14:41Z"},{"alias_kind":"pith_short_16","alias_value":"ECPHSAK344C3KIDE","created_at":"2026-07-05T06:14:41Z"},{"alias_kind":"pith_short_8","alias_value":"ECPHSAK3","created_at":"2026-07-05T06:14:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:ECPHSAK344C3KIDEQFVPC2BVUZ","target":"record","payload":{"canonical_record":{"source":{"id":"2303.05505","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-09T18:56:26Z","cross_cats_sorted":[],"title_canon_sha256":"2f72e865ee266799e729e6183e480fc9910bbffe3c42362ced34ac552f50e431","abstract_canon_sha256":"2e0b50e7ca89a4004c0ca007e7b7f4956c56ecba1a16a67420dc738368e3092d"},"schema_version":"1.0"},"canonical_sha256":"209e79015be705b52064816af16835a660b8982dd63a5fdb073b96358ff520b6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:14:41.600431Z","signature_b64":"G5GpZbDy7BTKXRZmUqYCYIG+V22OGxF6P1n7ujlLP9cyqj3KYKhlz9xHNKtcZYNoAACBay/Gx5MMcHQw8Q/zDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"209e79015be705b52064816af16835a660b8982dd63a5fdb073b96358ff520b6","last_reissued_at":"2026-07-05T06:14:41.599730Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:14:41.599730Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2303.05505","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:14:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ubn9b0l81XN2vQfwd7fVt4NFMynT7AupprmE+JdrpOIyY4IPVDyLdB65taqSfmR5pP0EcCn722I9kzLIAO33Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-09T00:59:28.175470Z"},"content_sha256":"08b085b7612fda40d0cf1c12dc3b3c41615c0f54b8bfc9deb2554dcd609b6f87","schema_version":"1.0","event_id":"sha256:08b085b7612fda40d0cf1c12dc3b3c41615c0f54b8bfc9deb2554dcd609b6f87"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:ECPHSAK344C3KIDEQFVPC2BVUZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On interval colourings of graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Julien Portier, Lawrence Hollom, Leo Versteegen","submitted_at":"2023-03-09T18:56:26Z","abstract_excerpt":"An interval colouring of a graph $G=(V,E)$ is a proper colouring $c\\colon E\\to \\mathbb{Z}$ such that the set of colours of edges incident to any given vertex forms an interval of $\\mathbb{Z}$. The interval thickness $\\theta(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be edge-partitioned into $k$ interval colourable graphs, and $\\theta(n)$ is the largest interval thickness over graphs on $n$ vertices. We show that $c \\frac{\\log n}{\\log \\log n} \\leq \\theta(n) \\leq n^{8/9+o(1)}$ for some $c>0$. In particular this answers a question by Asratian, Casselgren, and Petrosyan.\n  In"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.05505","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.05505/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T06:14:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qy/ZBZXCzWQreTlikViJpUBcPaQCgdnzTPLmhJCI/4lrlxQN/yW3oAgbCZNV/Imx+krdc6Fm1UA4ck5sYJEGAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-09T00:59:28.175858Z"},"content_sha256":"4e854fc899b4b7b3c92960b2acbdf3fbdfee37e607d56939bff87457079b3ecf","schema_version":"1.0","event_id":"sha256:4e854fc899b4b7b3c92960b2acbdf3fbdfee37e607d56939bff87457079b3ecf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ECPHSAK344C3KIDEQFVPC2BVUZ/bundle.json","state_url":"https://pith.science/pith/ECPHSAK344C3KIDEQFVPC2BVUZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ECPHSAK344C3KIDEQFVPC2BVUZ/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":"2026-07-09T00:59:28Z","links":{"resolver":"https://pith.science/pith/ECPHSAK344C3KIDEQFVPC2BVUZ","bundle":"https://pith.science/pith/ECPHSAK344C3KIDEQFVPC2BVUZ/bundle.json","state":"https://pith.science/pith/ECPHSAK344C3KIDEQFVPC2BVUZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ECPHSAK344C3KIDEQFVPC2BVUZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:ECPHSAK344C3KIDEQFVPC2BVUZ","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":"2e0b50e7ca89a4004c0ca007e7b7f4956c56ecba1a16a67420dc738368e3092d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-09T18:56:26Z","title_canon_sha256":"2f72e865ee266799e729e6183e480fc9910bbffe3c42362ced34ac552f50e431"},"schema_version":"1.0","source":{"id":"2303.05505","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.05505","created_at":"2026-07-05T06:14:41Z"},{"alias_kind":"arxiv_version","alias_value":"2303.05505v2","created_at":"2026-07-05T06:14:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.05505","created_at":"2026-07-05T06:14:41Z"},{"alias_kind":"pith_short_12","alias_value":"ECPHSAK344C3","created_at":"2026-07-05T06:14:41Z"},{"alias_kind":"pith_short_16","alias_value":"ECPHSAK344C3KIDE","created_at":"2026-07-05T06:14:41Z"},{"alias_kind":"pith_short_8","alias_value":"ECPHSAK3","created_at":"2026-07-05T06:14:41Z"}],"graph_snapshots":[{"event_id":"sha256:4e854fc899b4b7b3c92960b2acbdf3fbdfee37e607d56939bff87457079b3ecf","target":"graph","created_at":"2026-07-05T06:14:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2303.05505/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"An interval colouring of a graph $G=(V,E)$ is a proper colouring $c\\colon E\\to \\mathbb{Z}$ such that the set of colours of edges incident to any given vertex forms an interval of $\\mathbb{Z}$. The interval thickness $\\theta(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be edge-partitioned into $k$ interval colourable graphs, and $\\theta(n)$ is the largest interval thickness over graphs on $n$ vertices. We show that $c \\frac{\\log n}{\\log \\log n} \\leq \\theta(n) \\leq n^{8/9+o(1)}$ for some $c>0$. In particular this answers a question by Asratian, Casselgren, and Petrosyan.\n  In","authors_text":"Julien Portier, Lawrence Hollom, Leo Versteegen","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-09T18:56:26Z","title":"On interval colourings of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.05505","kind":"arxiv","version":2},"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":"sha256:08b085b7612fda40d0cf1c12dc3b3c41615c0f54b8bfc9deb2554dcd609b6f87","target":"record","created_at":"2026-07-05T06:14:41Z","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":"2e0b50e7ca89a4004c0ca007e7b7f4956c56ecba1a16a67420dc738368e3092d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-09T18:56:26Z","title_canon_sha256":"2f72e865ee266799e729e6183e480fc9910bbffe3c42362ced34ac552f50e431"},"schema_version":"1.0","source":{"id":"2303.05505","kind":"arxiv","version":2}},"canonical_sha256":"209e79015be705b52064816af16835a660b8982dd63a5fdb073b96358ff520b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"209e79015be705b52064816af16835a660b8982dd63a5fdb073b96358ff520b6","first_computed_at":"2026-07-05T06:14:41.599730Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:14:41.599730Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G5GpZbDy7BTKXRZmUqYCYIG+V22OGxF6P1n7ujlLP9cyqj3KYKhlz9xHNKtcZYNoAACBay/Gx5MMcHQw8Q/zDA==","signature_status":"signed_v1","signed_at":"2026-07-05T06:14:41.600431Z","signed_message":"canonical_sha256_bytes"},"source_id":"2303.05505","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:08b085b7612fda40d0cf1c12dc3b3c41615c0f54b8bfc9deb2554dcd609b6f87","sha256:4e854fc899b4b7b3c92960b2acbdf3fbdfee37e607d56939bff87457079b3ecf"],"state_sha256":"8b10d129950d955550522bfb00856a7974c42ca0a522e4187e41e162514272eb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cEEuh9p0pq9/0vxBH1fFq8Mn4STnJ3TgQ0SEgxigg8RyYK1BZYJbjtLxs1nkG62g0JGgM0rRZk1YL9MJpm0qDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-09T00:59:28.177831Z","bundle_sha256":"968524b501a539cf2e7869c0f3ca107f5ca0685fc5de863a2c7aafb57cf5b871"}}