The chromatic polynomial for cycle graphs
Pith reviewed 2026-05-25 00:09 UTC · model grok-4.3
The pith
The chromatic polynomial of the cycle graph C_n equals (λ-1)^n + (-1)^n(λ-1) for every n ≥ 1.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The chromatic polynomial P(C_n,λ) for the cycle graph C_n is P(C_n,λ)=(λ−1)^n+(−1)^n(λ−1) for all positive integers n≥1. The paper establishes this identity by presenting the standard inductive proof that relies on the deletion-contraction recurrence together with three additional proofs of the same closed form.
What carries the argument
The deletion-contraction recurrence P(G,λ)=P(G−e,λ)−P(G/e,λ) for any graph G and edge e, which supplies the inductive step for one of the four proofs.
Load-bearing premise
The deletion-contraction recurrence holds for the chromatic polynomial of every graph and every edge.
What would settle it
Direct enumeration of the proper 3-colorings of C_3 yields any number other than 6.
read the original abstract
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 $C_n$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript states the standard formula for the chromatic polynomial of the cycle graph C_n, namely P(C_n, λ) = (λ−1)^n + (−1)^n(λ−1) for n ≥ 1, and supplies four proofs of this identity: the familiar inductive argument via the deletion-contraction recurrence together with three additional derivations.
Significance. The identity is a textbook result whose correctness is already established in the literature. Re-deriving it by four routes may have modest pedagogical value in illustrating alternative proof strategies for chromatic polynomials, but the work introduces no new theorems, parameter-free derivations, or falsifiable predictions beyond the known closed form.
minor comments (3)
- The abstract and introduction should explicitly note that the formula and the deletion-contraction proof are standard (e.g., cite a textbook such as Bollobás or Diestel) so that the contribution is framed as a collection of alternative arguments rather than a new result.
- Section headings and proof labels should be numbered consistently (e.g., §2, §3, …) and each proof should end with a clear statement of what has been shown, to improve readability.
- The manuscript would benefit from a short concluding section that compares the four proofs (length, prerequisites, generality) rather than ending abruptly after the last argument.
Simulated Author's Rebuttal
We thank the referee for their review. The manuscript presents the standard formula for the chromatic polynomial of C_n along with four distinct proofs (one inductive via deletion-contraction and three others). We address the referee's observations below.
read point-by-point responses
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Referee: The identity is a textbook result whose correctness is already established in the literature. Re-deriving it by four routes may have modest pedagogical value in illustrating alternative proof strategies for chromatic polynomials, but the work introduces no new theorems, parameter-free derivations, or falsifiable predictions beyond the known closed form.
Authors: We agree that the closed-form expression is a standard textbook result. The manuscript's purpose is to collect and present four different derivations in one place, with three of them being non-inductive. While we do not claim a new theorem, we maintain that exhibiting multiple independent proofs can have pedagogical utility for readers interested in alternative techniques for chromatic polynomials, consistent with the referee's acknowledgment of modest pedagogical value. revision: no
Circularity Check
No significant circularity; standard result with independent proofs
full rationale
The paper restates the textbook formula for P(C_n, λ) and supplies four proofs, one of which applies the deletion-contraction recurrence (a general theorem for any graph, not an assumption or self-derived result). No equation, definition, or self-citation reduces the claimed identity to a fitted input, renamed ansatz, or prior result by the same authors. The derivation chain remains self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The deletion-contraction recurrence P(G,λ) = P(G−e,λ) − P(G/e,λ) holds for any graph G and edge e.
discussion (0)
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