{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:5NKUOT2UHEJWLFXP2ICR4DE2NE","short_pith_number":"pith:5NKUOT2U","canonical_record":{"source":{"id":"1606.00268","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2016-06-01T13:01:58Z","cross_cats_sorted":[],"title_canon_sha256":"e375edf7570c5aa512f6a3d2fdf007348c6001f51a70ed1be96d6568a7dd8802","abstract_canon_sha256":"fd3de3d47342d3117fc8c87f510aa999ce1d79aa6270d31398755e189260a5f9"},"schema_version":"1.0"},"canonical_sha256":"eb55474f5439136596efd2051e0c9a6913b302d0002343a8d58ec7f183111db6","source":{"kind":"arxiv","id":"1606.00268","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.00268","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"arxiv_version","alias_value":"1606.00268v1","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.00268","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"pith_short_12","alias_value":"5NKUOT2UHEJW","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5NKUOT2UHEJWLFXP","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5NKUOT2U","created_at":"2026-05-18T12:30:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:5NKUOT2UHEJWLFXP2ICR4DE2NE","target":"record","payload":{"canonical_record":{"source":{"id":"1606.00268","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2016-06-01T13:01:58Z","cross_cats_sorted":[],"title_canon_sha256":"e375edf7570c5aa512f6a3d2fdf007348c6001f51a70ed1be96d6568a7dd8802","abstract_canon_sha256":"fd3de3d47342d3117fc8c87f510aa999ce1d79aa6270d31398755e189260a5f9"},"schema_version":"1.0"},"canonical_sha256":"eb55474f5439136596efd2051e0c9a6913b302d0002343a8d58ec7f183111db6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:05.621504Z","signature_b64":"hAIvpFnCnHErTTUtUZkl3PzgURAZaeTP/m0EoTdJNZDxcd4LN19VWKtesuhfMruzvtUvmj2JqmfXPHX/9uGJBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb55474f5439136596efd2051e0c9a6913b302d0002343a8d58ec7f183111db6","last_reissued_at":"2026-05-18T01:13:05.621103Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:05.621103Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.00268","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":"2026-05-18T01:13:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NJSoiDW9C1Wr7z5TVIovuUIEh2xdzbSC9Rnl/9x7rvpAkVRPTN2H6G8ePrtSsWZUNNi0mHu/qt3i5S3RUar1Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T22:11:38.739820Z"},"content_sha256":"9953e7c6d0cbf76a8b6f211b4167ee0fe847e73b4dbf1931f8a738e4bf60b1a4","schema_version":"1.0","event_id":"sha256:9953e7c6d0cbf76a8b6f211b4167ee0fe847e73b4dbf1931f8a738e4bf60b1a4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:5NKUOT2UHEJWLFXP2ICR4DE2NE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Certain Chromatic Sums of Some Cycle Related Graph Classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Johan Kok, K. P. Chithra, N. K. Sudev","submitted_at":"2016-06-01T13:01:58Z","abstract_excerpt":"Let $\\mathcal{C} = \\{c_1,c_2, c_3, \\ldots,c_k\\}$ be a certain type of proper $k$-colouring of a given graph $G$ and $\\theta(c_i)$ denote the number of times a particular colour $c_i$ is assigned to the vertices of $G$. Then, the colouring sum of a given graph $G$ with respect to the colouring $\\cC$, denoted by $\\omega_{\\cC}(G)$, is defined to be $\\omega(\\cC) = \\sum\\limits_{i=1}^{k}i\\,\\theta(c_i)$. The colouring sums such as $\\chi$-chromatic sum, $\\chi^+$-chromatic sum, $b$-chromatic sum, $b^+$-chromatic sum etc. are some of these types of colouring sums that have been studied recently. Motivat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.00268","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":"2026-05-18T01:13:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i/k36yOPjNuZD8njafb7ZNsiOXwzq/NdxiS0flhp5Kk2ej1/+y9FV6jbyl6fBlxoL69Qt6sEdYrqzpP97xYRAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T22:11:38.740168Z"},"content_sha256":"c396e9edff0525ac8cc25830a8930abb7e8e85ffc5e1e2511029432be1b46191","schema_version":"1.0","event_id":"sha256:c396e9edff0525ac8cc25830a8930abb7e8e85ffc5e1e2511029432be1b46191"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5NKUOT2UHEJWLFXP2ICR4DE2NE/bundle.json","state_url":"https://pith.science/pith/5NKUOT2UHEJWLFXP2ICR4DE2NE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5NKUOT2UHEJWLFXP2ICR4DE2NE/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":"2026-06-03T22:11:38Z","links":{"resolver":"https://pith.science/pith/5NKUOT2UHEJWLFXP2ICR4DE2NE","bundle":"https://pith.science/pith/5NKUOT2UHEJWLFXP2ICR4DE2NE/bundle.json","state":"https://pith.science/pith/5NKUOT2UHEJWLFXP2ICR4DE2NE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5NKUOT2UHEJWLFXP2ICR4DE2NE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:5NKUOT2UHEJWLFXP2ICR4DE2NE","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":"fd3de3d47342d3117fc8c87f510aa999ce1d79aa6270d31398755e189260a5f9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2016-06-01T13:01:58Z","title_canon_sha256":"e375edf7570c5aa512f6a3d2fdf007348c6001f51a70ed1be96d6568a7dd8802"},"schema_version":"1.0","source":{"id":"1606.00268","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.00268","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"arxiv_version","alias_value":"1606.00268v1","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.00268","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"pith_short_12","alias_value":"5NKUOT2UHEJW","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5NKUOT2UHEJWLFXP","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5NKUOT2U","created_at":"2026-05-18T12:30:01Z"}],"graph_snapshots":[{"event_id":"sha256:c396e9edff0525ac8cc25830a8930abb7e8e85ffc5e1e2511029432be1b46191","target":"graph","created_at":"2026-05-18T01:13:05Z","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":"Let $\\mathcal{C} = \\{c_1,c_2, c_3, \\ldots,c_k\\}$ be a certain type of proper $k$-colouring of a given graph $G$ and $\\theta(c_i)$ denote the number of times a particular colour $c_i$ is assigned to the vertices of $G$. Then, the colouring sum of a given graph $G$ with respect to the colouring $\\cC$, denoted by $\\omega_{\\cC}(G)$, is defined to be $\\omega(\\cC) = \\sum\\limits_{i=1}^{k}i\\,\\theta(c_i)$. The colouring sums such as $\\chi$-chromatic sum, $\\chi^+$-chromatic sum, $b$-chromatic sum, $b^+$-chromatic sum etc. are some of these types of colouring sums that have been studied recently. Motivat","authors_text":"Johan Kok, K. P. Chithra, N. K. Sudev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2016-06-01T13:01:58Z","title":"Certain Chromatic Sums of Some Cycle Related Graph Classes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.00268","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":"sha256:9953e7c6d0cbf76a8b6f211b4167ee0fe847e73b4dbf1931f8a738e4bf60b1a4","target":"record","created_at":"2026-05-18T01:13:05Z","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":"fd3de3d47342d3117fc8c87f510aa999ce1d79aa6270d31398755e189260a5f9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2016-06-01T13:01:58Z","title_canon_sha256":"e375edf7570c5aa512f6a3d2fdf007348c6001f51a70ed1be96d6568a7dd8802"},"schema_version":"1.0","source":{"id":"1606.00268","kind":"arxiv","version":1}},"canonical_sha256":"eb55474f5439136596efd2051e0c9a6913b302d0002343a8d58ec7f183111db6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eb55474f5439136596efd2051e0c9a6913b302d0002343a8d58ec7f183111db6","first_computed_at":"2026-05-18T01:13:05.621103Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:13:05.621103Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hAIvpFnCnHErTTUtUZkl3PzgURAZaeTP/m0EoTdJNZDxcd4LN19VWKtesuhfMruzvtUvmj2JqmfXPHX/9uGJBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:13:05.621504Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.00268","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9953e7c6d0cbf76a8b6f211b4167ee0fe847e73b4dbf1931f8a738e4bf60b1a4","sha256:c396e9edff0525ac8cc25830a8930abb7e8e85ffc5e1e2511029432be1b46191"],"state_sha256":"bfd6dd59469698636d77ffec6a8a7e9d721f74ae2c05355089854b134b8c463f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XYcX6JqgGcTLk/3l2FyjlrHCwh3KodE93CUioQ5k0XNlYAaRaJvl3YZaV7filHO2XB1fhKNrQIlPNyVEaDv0Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T22:11:38.742161Z","bundle_sha256":"4fd296e3c95057394ccf2cef0754e17149b6ad5e70701e5beffee3c7e7d2a5d5"}}