AI 中文总结
该研究证明了Ghosh等人关于平面图中偶长度路径最大数量的猜想，同时解决了Cox–Martin优化猜想。
AI 中文摘要
对于图\textit{G}和\textit{H}，设\textit{N(G,H)}为\textit{G}中未标记、不一定是诱导子图的\textit{H}副本数量，\textit{f(n,H)}为所有\textit{n}顶点平面图\textit{G}中\textit{N(G,H)}的最大值。Ghosh等人猜想，对每个固定整数\textit{ℓ≥2}，\textit{f(n,P_{2ℓ+1})=4ℓ(n/ℓ)^{ℓ+1}+O(n^ℓ)}，我们证明了该猜想及所述误差项，同时还解决了Cox–Martin优化猜想。
英文摘要
For graphs \(G\) and \(H\), let \(N(G,H)\) be the number of unlabeled, not necessarily induced copies of \(H\) in \(G\), and let \(f(n,H)\) be the maximum of \(N(G,H)\) over all \(n\)-vertex planar graphs \(G\). Ghosh, Győri, Martin, Paulos, Salia, Xiao and Zamora conjectured that, for every fixed integer \(\ell\ge 2\), \[ f(n,P_{2\ell+1}) =4\ell\left(\frac{n}{\ell}\right)^{\ell+1}+O(n^\ell). \] We prove the conjecture, including the stated error term. Along the way, we also settle the Cox--Martin optimization conjecture.
Comments9 pages
