{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:SRN2SFZNZTME3RU4XFYXNFDR3I","short_pith_number":"pith:SRN2SFZN","canonical_record":{"source":{"id":"1907.04320","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2019-07-09T07:37:34Z","cross_cats_sorted":[],"title_canon_sha256":"fe413519e8828ba3cb00b8ad5716d52cabe0fa3d9c39aa60419358f29d6ec7a8","abstract_canon_sha256":"609b214f4e34f9b41239cc1fb0cb1495ef3920b0c6955f0fc4f75414aeb6f90c"},"schema_version":"1.0"},"canonical_sha256":"945ba9172dccd84dc69cb971769471da0a36c4a558102e93e34617cf9eeb11a5","source":{"kind":"arxiv","id":"1907.04320","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.04320","created_at":"2026-05-17T23:41:02Z"},{"alias_kind":"arxiv_version","alias_value":"1907.04320v1","created_at":"2026-05-17T23:41:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.04320","created_at":"2026-05-17T23:41:02Z"},{"alias_kind":"pith_short_12","alias_value":"SRN2SFZNZTME","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"SRN2SFZNZTME3RU4","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"SRN2SFZN","created_at":"2026-05-18T12:33:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:SRN2SFZNZTME3RU4XFYXNFDR3I","target":"record","payload":{"canonical_record":{"source":{"id":"1907.04320","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2019-07-09T07:37:34Z","cross_cats_sorted":[],"title_canon_sha256":"fe413519e8828ba3cb00b8ad5716d52cabe0fa3d9c39aa60419358f29d6ec7a8","abstract_canon_sha256":"609b214f4e34f9b41239cc1fb0cb1495ef3920b0c6955f0fc4f75414aeb6f90c"},"schema_version":"1.0"},"canonical_sha256":"945ba9172dccd84dc69cb971769471da0a36c4a558102e93e34617cf9eeb11a5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:02.087771Z","signature_b64":"mHdxNRR2e+USBKvx9gXhXghYMkHXSzt4LusEhO9hoLUGhaZ23ZgYmOZVTuFaEaGxGS6ZKO6wwPdqmcKh59xvDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"945ba9172dccd84dc69cb971769471da0a36c4a558102e93e34617cf9eeb11a5","last_reissued_at":"2026-05-17T23:41:02.087035Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:02.087035Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.04320","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":"2026-05-17T23:41:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4uDehTDJkjvueeOAr20mP/ydDe1Jay0mbiSHzPG5yQu5NVJ1OUikRKpEwrqU+XB10ZB1JVpTjRaT9Hxf0eFmCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T01:50:34.387718Z"},"content_sha256":"b19d5107c2de3497053096c4c178a9eb954667ee02039d96e85b9a00501a5d57","schema_version":"1.0","event_id":"sha256:b19d5107c2de3497053096c4c178a9eb954667ee02039d96e85b9a00501a5d57"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:SRN2SFZNZTME3RU4XFYXNFDR3I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The chromatic polynomial for cycle graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Heesung Shin, Jonghyeon Lee","submitted_at":"2019-07-09T07:37:34Z","abstract_excerpt":"Let $P(G,\\lambda)$ denote the number of proper vertex colorings of $G$ with $\\lambda$ colors. The chromatic polynomial $P(C_n,\\lambda)$ for the cycle graph $C_n$ is well-known as $$P(C_n,\\lambda) = (\\lambda-1)^n+(-1)^n(\\lambda-1)$$ for all positive integers $n\\ge 1$. Also its inductive proof is widely well-known by the \\emph{deletion-contraction recurrence}. In this paper, we give this inductive proof again and three other proofs of this formula of the chromatic polynomial for the cycle graph $C_n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04320","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":"2026-05-17T23:41:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jTKq03TAX9Lpw+rOINSnsx0DxQchTF52iZax/esGb71c5kbhdbWL4MEFcCglgPE5IhIiqEkvIWCcLWzLuWzXCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T01:50:34.388400Z"},"content_sha256":"fbef5c8ac10a743bd4c95b7f80d0bdbf05b828d37cc7463dc72b7d782a392dba","schema_version":"1.0","event_id":"sha256:fbef5c8ac10a743bd4c95b7f80d0bdbf05b828d37cc7463dc72b7d782a392dba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SRN2SFZNZTME3RU4XFYXNFDR3I/bundle.json","state_url":"https://pith.science/pith/SRN2SFZNZTME3RU4XFYXNFDR3I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SRN2SFZNZTME3RU4XFYXNFDR3I/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":"2026-05-26T01:50:34Z","links":{"resolver":"https://pith.science/pith/SRN2SFZNZTME3RU4XFYXNFDR3I","bundle":"https://pith.science/pith/SRN2SFZNZTME3RU4XFYXNFDR3I/bundle.json","state":"https://pith.science/pith/SRN2SFZNZTME3RU4XFYXNFDR3I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SRN2SFZNZTME3RU4XFYXNFDR3I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:SRN2SFZNZTME3RU4XFYXNFDR3I","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":"609b214f4e34f9b41239cc1fb0cb1495ef3920b0c6955f0fc4f75414aeb6f90c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2019-07-09T07:37:34Z","title_canon_sha256":"fe413519e8828ba3cb00b8ad5716d52cabe0fa3d9c39aa60419358f29d6ec7a8"},"schema_version":"1.0","source":{"id":"1907.04320","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.04320","created_at":"2026-05-17T23:41:02Z"},{"alias_kind":"arxiv_version","alias_value":"1907.04320v1","created_at":"2026-05-17T23:41:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.04320","created_at":"2026-05-17T23:41:02Z"},{"alias_kind":"pith_short_12","alias_value":"SRN2SFZNZTME","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"SRN2SFZNZTME3RU4","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"SRN2SFZN","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:fbef5c8ac10a743bd4c95b7f80d0bdbf05b828d37cc7463dc72b7d782a392dba","target":"graph","created_at":"2026-05-17T23:41:02Z","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 $P(G,\\lambda)$ denote the number of proper vertex colorings of $G$ with $\\lambda$ colors. The chromatic polynomial $P(C_n,\\lambda)$ for the cycle graph $C_n$ is well-known as $$P(C_n,\\lambda) = (\\lambda-1)^n+(-1)^n(\\lambda-1)$$ for all positive integers $n\\ge 1$. Also its inductive proof is widely well-known by the \\emph{deletion-contraction recurrence}. In this paper, we give this inductive proof again and three other proofs of this formula of the chromatic polynomial for the cycle graph $C_n$.","authors_text":"Heesung Shin, Jonghyeon Lee","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2019-07-09T07:37:34Z","title":"The chromatic polynomial for cycle graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04320","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:b19d5107c2de3497053096c4c178a9eb954667ee02039d96e85b9a00501a5d57","target":"record","created_at":"2026-05-17T23:41:02Z","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":"609b214f4e34f9b41239cc1fb0cb1495ef3920b0c6955f0fc4f75414aeb6f90c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2019-07-09T07:37:34Z","title_canon_sha256":"fe413519e8828ba3cb00b8ad5716d52cabe0fa3d9c39aa60419358f29d6ec7a8"},"schema_version":"1.0","source":{"id":"1907.04320","kind":"arxiv","version":1}},"canonical_sha256":"945ba9172dccd84dc69cb971769471da0a36c4a558102e93e34617cf9eeb11a5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"945ba9172dccd84dc69cb971769471da0a36c4a558102e93e34617cf9eeb11a5","first_computed_at":"2026-05-17T23:41:02.087035Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:02.087035Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mHdxNRR2e+USBKvx9gXhXghYMkHXSzt4LusEhO9hoLUGhaZ23ZgYmOZVTuFaEaGxGS6ZKO6wwPdqmcKh59xvDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:02.087771Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.04320","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b19d5107c2de3497053096c4c178a9eb954667ee02039d96e85b9a00501a5d57","sha256:fbef5c8ac10a743bd4c95b7f80d0bdbf05b828d37cc7463dc72b7d782a392dba"],"state_sha256":"e91105ed41293649ddf0777c9d37a396fad20dc4874005f951fc9ed54d37efee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7BJTt1I/ruU1EVwO6CDr1sp3UNTdetUyNijEyKRvW1x0+bgzgA350B5JBQs5ZRJnuZjs7d7m0L95TezziQ/GDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T01:50:34.392133Z","bundle_sha256":"53eb13e907ec31897e7cdd6e903cff2aabc702bf925746f7ace12ce2fbeef7b7"}}