一类含花刺的三次图
On a family of cubic graphs containing the flower snarks
摘要：
我们考虑由$k \geq 2$不相交爪$C_i \sim K_{1, 3}$（$0 \leq i \leq k-1$）形成的三次图，使得对于每个整数$i$模$k$，$\ C_i$的三个1度顶点连接到$C_{i-1}$的三个1度顶点，并连接到$C_{i+1}$的三个1度顶点.用$t_i$表示$C_i$的3次顶点，用$T$表示集合$\{t_1, t_2,..., t_{k-1}\}$。在这样的方式中，我们构造三个不同的图，即$FS(1,k)$，$FS(2,k)$和$FS(3,k)$。图$FS(j,k)$（$j \in \{1, 2, 3\}$）是其中顶点集$\cup_{i=0}^{i=k-1}V(C_i) \setminus T$诱导$j$循环的图（注意，图$FS(2,2p+1)$、$p\geq2$是由Isaacs \cite{Isa75}定义的花蛇）。我们确定每个$FS(j,k)$的完美匹配的数量。一个三次图$G$称为{$G$ 2-因子哈密尔顿}，如果$G$的每一个2-因子都是哈密尔顿圈。我们刻画了2-因子Hamilton图$FS(j,k)$（注意FS（1，3）是Robertson，Seymour和Thomas \cite{RobSey}的“三重图”）.图$G$中的{ \em强匹配} $M$是匹配$M$，使得没有$E(G)$的边连接$M$的任何两个边。具有两个强匹配的完美匹配并的三次图称为{\Jaev }。我们刻画了图$FS(j,k)$是\Jaesv。
英文摘要：
We consider cubic graphs formed with $k \geq 2$ disjoint claws $C_i \sim K_{1, 3}$ ($0 \leq i \leq k-1$) such that for every integer $i$ modulo $k$ the three vertices of degree 1 of $\ C_i$ are joined to the three vertices of degree 1 of $C_{i-1}$ and joined to the three vertices of degree 1 of $C_{i+1}$. Denote by $t_i$ the vertex of degree 3 of $C_i$ and by $T$ the set $\{t_1, t_2,..., t_{k-1}\}$. In such a way we construct three distinct graphs, namely $FS(1,k)$, $FS(2,k)$ and $FS(3,k)$. The graph $FS(j,k)$ ($j \in \{1, 2, 3\}$) is the graph where the set of vertices $\cup_{i=0}^{i=k-1}V(C_i) \setminus T$ induce $j$ cycles (note that the graphs $FS(2,2p+1)$, $p\geq2$, are the flower snarks defined by Isaacs \cite{Isa75}). We determine the number of perfect matchings of every $FS(j,k)$. A cubic graph $G$ is said to be {\em 2-factor hamiltonian} if every 2-factor of $G$ is a hamiltonian cycle. We characterize the graphs $FS(j,k)$ that are 2-factor hamiltonian (note that FS(1,3) is the "Triplex Graph" of Robertson, Seymour and Thomas \cite{RobSey}). A {\em strong matching} $M$ in a graph $G$ is a matching $M$ such that there is no edge of $E(G)$ connecting any two edges of $M$. A cubic graph having a perfect matching union of two strong matchings is said to be a {\em\Jaev}. We characterize the graphs $FS(j,k)$ that are \Jaesv.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
