AI 中文总结
本文研究了具有受限周长的图中奇环计数问题，证明了特定条件下图中奇环的最大数量，并解决了相关猜想。
AI 中文摘要
对于整数\(L\geq2\)，设\(a = \lfloor L/2\rfloor\)。令\(H(n,L)\)为\(K_a\)与一个阶为\(n - a\)的独立集的联图，当\(L\)为奇数时在独立集中有一条额外边。证明了对于每个固定的\(s\geq3\)和\(L\geq2s + 2\)以及所有足够大的\(n\)，\(\operatorname{ex}(n,C_{2s + 1},\mathcal{C}_{\geq L + 1}) = N(C_{2s + 1},H(n,L))\)。结合赵和王关于偶圈的近期结果，解决了朱等人关于在有界周长图中计算固定圈的猜想。还确定了禁止路径时奇圈的相应最大数量。
英文摘要
For an integer $L\ge2$, let $a=\lfloor L/2\rfloor$. Let $H(n,L)$ be the join of $K_a$ and an independent set of order $n-a$, with one extra edge in the independent set when $L$ is odd. We prove that, for fixed integers $q\ge4$ and $L>q$, and for all sufficiently large $n$, the graph $H(n,L)$ maximizes the number of copies of $C_q$ among all $n$-vertex graphs of circumference at most $L$. This settles a conjecture of Zhu, Győri, He, Lv, Salia and Xiao~[Bull. Lond. Math. Soc. 55 (2023)]. For even $q\ge6$, we also prove the boundary case $L=q$. We further determine the corresponding maximum when a long path is forbidden.
