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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 $C_n$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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. 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