A bond in a graph is a minimal nonempty edge-cut. A connected graph $G$ is dual Hamiltonian if the vertex set can be partitioned into two subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are both trees. There is much interest in studying the longest cycles and largest bonds in graphs. H. Wu conjectured that any longest cycle must meet any largest bond in a simple 3-connected graph. In this paper, the author proves that the above conjecture is true for certain classes of 3-connected graphs: Let $G$ be a simple 3-connected graph with $n$ vertices and $m$ edges. Suppose $c(G)$ is the size of a longest cycle, and $c^*(G)$ is the size of a largest bond. Then each longest cycle meets each largest bond if either $c(G) \geq n - 3$ or $c^*(G) \geq m - n - 1$. Sanford determined in her Ph.D. thesis the cycle spectrum of the well-known generalized Petersen graph $P(n, 2)$ ($n$ is odd) and $P(n, 3)$ ($n$ is even). Flynn proved in her honors thesis that any generalized Petersen graph $P(n, k)$ is dual Hamiltonian. The author studies the bond spectrum (called the co-spectrum) of the generalized Petersen graphs and extends Flynn's result by proving that in any generalized Petersen graph $P(n, k)$, $1 \leq k < \frac{n}{2}$, the co-spectrum of $P(n, k)$ is $\{3, 4, 5, ..., n+2\}$.

翻译：图中的一个键是最小非空边割。若连通图$G$的顶点集可划分为两个子集$X$和$Y$，使得$X$和$Y$诱导的子图均为树，则称$G$为对偶哈密顿图。研究图中的最长圈和最大键具有重要意义。吴浩提出猜想：在简单3-连通图中，任意最长圈必与任意最大键相交。本文证明该猜想对某些3-连通图类成立：设$G$为具有$n$个顶点和$m$条边的简单3-连通图，$c(G)$为最长圈的长度，$c^*(G)$为最大键的大小。若$c(G) \geq n - 3$或$c^*(G) \geq m - n - 1$，则每个最长圈与每个最大键均相交。Sanford在其博士论文中确定了著名广义彼得森图$P(n, 2)$（$n$为奇数）和$P(n, 3)$（$n$为偶数）的圈谱。Flynn在其荣誉学士论文中证明任意广义彼得森图$P(n, k)$均为对偶哈密顿图。本文研究广义彼得森图的键谱（称为余谱），并推广Flynn的结果：证明对于任意广义彼得森图$P(n, k)$（$1 \leq k < \frac{n}{2}$），其键谱为$\{3, 4, 5, ..., n+2\}$。