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Counting Fixed Cycles in Graphs with Bounded Circumference

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bounded-circumference-cycle-countingCombinatoricsposed by Zhu, Gyori, He, Lv, Salia, Xiao, 2023recorded: solved

1 attempt · no person has looked

Statement

Zhu, Gyori, He, Lv, Salia and Xiao conjectured the maximum number of copies of a fixed cycle in an nn-vertex graph of bounded circumference, attained by the join of a clique with an independent set. For every fixed s≥3s \ge 3 and L≥2s+2L \ge 2s+2 and all large nn, ex(n,C2s+1,C≥L+1)=N(C2s+1,H(n,L))\mathrm{ex}(n, C_{2s+1}, \mathcal{C}_{\ge L+1}) = N(C_{2s+1}, H(n,L)). Together with the companion even-cycle result this settles the conjecture.

Context

the odd-cycle half; the even-cycle half is a companion paper by the same authors

A 2023 Bulletin of the LMS conjecture in generalized Turan theory, real and cited but recent and specialized.

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  • #1

    Attempt 1

    proof attemptGPT-5.6 with Xiamiao Zhao, Yuanpei Wang ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    GPT-5.6
    people
    Xiamiao Zhao, Yuanpei Wang

    The declaration credits the model with solving one case of Theorem 1.2, in particular the calculations in that proof, and with rewriting the Section 2.4 argument in the language of directed graphs; the rest is readability and exposition. The authors reviewed and verified the proofs and take sole responsibility.

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