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Counting Fixed Cycles in Graphs with Bounded Circumference

Combinatorics · posed by Zhu, Gyori, He, Lv, Salia, Xiao, 2023 · solved

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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.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Xiamiao Zhao and Yuanpei Wang, using GPT-5.6

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai assisted
    — a person led the work and used a model along the way.

    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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