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Off-Diagonal Commonality of a Shorter Even Cycle and a Longer Odd Cycle

Primarily AI-generated textHuman understanding: all partsmath.CO — Combinatorics

Contributed by Taeyoung Kim ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

Let n≥4n\ge4 be even and let ℓ>n\ell>n be odd. We determine the complete off-diagonal commonality region of the ordered pair (Cn,Cℓ)(C_n,C_\ell): it is the half-open interval (0,an,ℓ⋆](0,a_{n,\ell}^{\star}] cut out by the unique root an,ℓ⋆∈(1/2,1)a_{n,\ell}^{\star}\in(1/2,1) of an−1(n+(ℓ−n)a)ℓ=21−n.\frac{a^{n-1}\bigl(n+(\ell-n)a\bigr)}{\ell}=2^{1-n}. The upper endpoint is witnessed by the balanced disjoint union of two cliques. For the lower bound, cycle densities are expressed as traces of graphon operators. Rank-one eigenvalue majorisation separates the Perron eigenvalue from the rest of the spectrum, while the Hilbert--Schmidt norm leaves room for at most one negative eigenvalue to contribute with the wrong sign. That eigenvalue is paid for by the Perron eigenvalue, leaving the elementary inequality xn+(1−x)n≥21−nx^{n}+(1-x)^{n}\ge2^{1-n}.

Provenance statement

The proof was obtained by one-shot by ChatGPT 5.5 Pro with specific proof direction included in the prompt.

Tools used

OpenAI
ChatGPTVersion 5.5 Pro

References

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Version history

  1. v1 · 11 pagesSubmitted by Taeyoung KimInitial depositCurrentNov 03, 2026