{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:HE4XTG4WSTFS3PUTDC2LAX2P2Y","short_pith_number":"pith:HE4XTG4W","canonical_record":{"source":{"id":"1509.00275","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-01T13:24:18Z","cross_cats_sorted":[],"title_canon_sha256":"7d4f21147c3fc949309fb4d84a18908711ae764b0f139ca3c4620e0fc5839e83","abstract_canon_sha256":"06dfc1bf27fc559063f4ac7143827991a1b484922c37f243c81ddfeb16169142"},"schema_version":"1.0"},"canonical_sha256":"3939799b9694cb2dbe9318b4b05f4fd602f2f6d51baf47aa4d4ac35193f87734","source":{"kind":"arxiv","id":"1509.00275","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.00275","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"arxiv_version","alias_value":"1509.00275v1","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.00275","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"pith_short_12","alias_value":"HE4XTG4WSTFS","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"HE4XTG4WSTFS3PUT","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"HE4XTG4W","created_at":"2026-05-18T12:29:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:HE4XTG4WSTFS3PUTDC2LAX2P2Y","target":"record","payload":{"canonical_record":{"source":{"id":"1509.00275","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-01T13:24:18Z","cross_cats_sorted":[],"title_canon_sha256":"7d4f21147c3fc949309fb4d84a18908711ae764b0f139ca3c4620e0fc5839e83","abstract_canon_sha256":"06dfc1bf27fc559063f4ac7143827991a1b484922c37f243c81ddfeb16169142"},"schema_version":"1.0"},"canonical_sha256":"3939799b9694cb2dbe9318b4b05f4fd602f2f6d51baf47aa4d4ac35193f87734","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:14.590450Z","signature_b64":"hocxtdyxU4XaXdFXOiCKBkzZel6ab3uD6cByWH65yxtobbAi/a7opygr27pPJV7w7tlUug70o2xu29hxMvG3DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3939799b9694cb2dbe9318b4b05f4fd602f2f6d51baf47aa4d4ac35193f87734","last_reissued_at":"2026-05-18T01:34:14.589621Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:14.589621Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.00275","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":"2026-05-18T01:34:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gZrs9OBDK16I9eLnP67KKc3i1T2kl6BqJcWlT0ECVe7ErqXNot4jZa9NxtBSpkDSGerrBmU4EX1tOayNICP1AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T23:10:04.598857Z"},"content_sha256":"16b9240fe9062bacc196b7904e0049f8f4c1d27dea23dd9c48b9135051386dbd","schema_version":"1.0","event_id":"sha256:16b9240fe9062bacc196b7904e0049f8f4c1d27dea23dd9c48b9135051386dbd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:HE4XTG4WSTFS3PUTDC2LAX2P2Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A characterization of L(2, 1)-labeling number for trees with maximum degree 3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dong Chen, Pak Kiu Sun, Qiaojun Shu, Wai Chee Shiu, Weifan Wang","submitted_at":"2015-09-01T13:24:18Z","abstract_excerpt":"An L(2, 1)-labeling of a graph is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive numbers differed by at least 2, and vertices at distance 2 are assigned distinct numbers. The L(2, 1)-labeling number is the minimum range of labels over all such labeling. It was shown by Griggs and Yeh [Labelling graphs with a condition at distance 2, SIAM J. Discrete Math. 5(1992), 586-595] that the L(2, 1)-labeling number of a tree is either \\D+ 1 or \\D + 2. In this paper, we give a complete characterization of L(2, 1)-labeling number for trees with maximum degre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00275","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":"2026-05-18T01:34:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wVYRDZv1IFT5OuX7s64AHMJddtV1VG/LGb4ajQMNlzFCy601+oSCm5+Ij7XntTTN6VxafW/IqGuw86CX4ip3BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T23:10:04.599662Z"},"content_sha256":"a4dbe920c2c5c367c1cf308a4748ff5b448b2ec1795be28849780e2fb9b2e881","schema_version":"1.0","event_id":"sha256:a4dbe920c2c5c367c1cf308a4748ff5b448b2ec1795be28849780e2fb9b2e881"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HE4XTG4WSTFS3PUTDC2LAX2P2Y/bundle.json","state_url":"https://pith.science/pith/HE4XTG4WSTFS3PUTDC2LAX2P2Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HE4XTG4WSTFS3PUTDC2LAX2P2Y/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":"2026-05-30T23:10:04Z","links":{"resolver":"https://pith.science/pith/HE4XTG4WSTFS3PUTDC2LAX2P2Y","bundle":"https://pith.science/pith/HE4XTG4WSTFS3PUTDC2LAX2P2Y/bundle.json","state":"https://pith.science/pith/HE4XTG4WSTFS3PUTDC2LAX2P2Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HE4XTG4WSTFS3PUTDC2LAX2P2Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:HE4XTG4WSTFS3PUTDC2LAX2P2Y","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":"06dfc1bf27fc559063f4ac7143827991a1b484922c37f243c81ddfeb16169142","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-01T13:24:18Z","title_canon_sha256":"7d4f21147c3fc949309fb4d84a18908711ae764b0f139ca3c4620e0fc5839e83"},"schema_version":"1.0","source":{"id":"1509.00275","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.00275","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"arxiv_version","alias_value":"1509.00275v1","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.00275","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"pith_short_12","alias_value":"HE4XTG4WSTFS","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"HE4XTG4WSTFS3PUT","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"HE4XTG4W","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:a4dbe920c2c5c367c1cf308a4748ff5b448b2ec1795be28849780e2fb9b2e881","target":"graph","created_at":"2026-05-18T01:34:14Z","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":"An L(2, 1)-labeling of a graph is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive numbers differed by at least 2, and vertices at distance 2 are assigned distinct numbers. The L(2, 1)-labeling number is the minimum range of labels over all such labeling. It was shown by Griggs and Yeh [Labelling graphs with a condition at distance 2, SIAM J. Discrete Math. 5(1992), 586-595] that the L(2, 1)-labeling number of a tree is either \\D+ 1 or \\D + 2. In this paper, we give a complete characterization of L(2, 1)-labeling number for trees with maximum degre","authors_text":"Dong Chen, Pak Kiu Sun, Qiaojun Shu, Wai Chee Shiu, Weifan Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-01T13:24:18Z","title":"A characterization of L(2, 1)-labeling number for trees with maximum degree 3"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00275","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":"sha256:16b9240fe9062bacc196b7904e0049f8f4c1d27dea23dd9c48b9135051386dbd","target":"record","created_at":"2026-05-18T01:34:14Z","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":"06dfc1bf27fc559063f4ac7143827991a1b484922c37f243c81ddfeb16169142","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-01T13:24:18Z","title_canon_sha256":"7d4f21147c3fc949309fb4d84a18908711ae764b0f139ca3c4620e0fc5839e83"},"schema_version":"1.0","source":{"id":"1509.00275","kind":"arxiv","version":1}},"canonical_sha256":"3939799b9694cb2dbe9318b4b05f4fd602f2f6d51baf47aa4d4ac35193f87734","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3939799b9694cb2dbe9318b4b05f4fd602f2f6d51baf47aa4d4ac35193f87734","first_computed_at":"2026-05-18T01:34:14.589621Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:14.589621Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hocxtdyxU4XaXdFXOiCKBkzZel6ab3uD6cByWH65yxtobbAi/a7opygr27pPJV7w7tlUug70o2xu29hxMvG3DA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:14.590450Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.00275","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:16b9240fe9062bacc196b7904e0049f8f4c1d27dea23dd9c48b9135051386dbd","sha256:a4dbe920c2c5c367c1cf308a4748ff5b448b2ec1795be28849780e2fb9b2e881"],"state_sha256":"485a09fea81defe6e6c3253534981da0bc90e756a3ce3d171938d6236191ab26"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rMRYB3czy41wWqgUMSgJGKGT5vxYarLg2RvO+ODwysKdqVfQBX0IuxyGWmfF2QEI8vQiPdZh2FVA7j9w+BpmAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T23:10:04.603971Z","bundle_sha256":"28827ad260417fcae3bc770305c74e31111b429f0b450207b3bbe1e3a0020891"}}