VibeMathedMath problems solved by AI

Counting Fixed Cycles in Graphs with Bounded Circumference

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.

Result
Proved(see note)
Status
Resolved
AI contribution
AI-assisted
Method
Argument
Field
Extremal graph theory
Posed by
Zhu, Gyori, He, Lv, Salia, Xiao
Year posed
2023
Years open
3y
Solved
2026-07-12
Model
GPT-5.6
Vendor
OpenAI
Collaborators
Xiamiao Zhao, Yuanpei Wang
Verification
Unreviewed
Publication
Preprint
Significance
10 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

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

What the AI did

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.

Verification

arXiv preprint; not yet peer-reviewed.

Source

arXiv:2607.10779 - Counting Odd Cycles in Graphs with Bounded Circumference

Discussion