{"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"}