直积图的超限制边连通性
The super restricted edge-connectedness of direct product graphs
摘要：
设$G$是一个顶点集为$V(G)$，边集为$E(G)$的图.如果$G-F$是不连通的并且没有孤立的顶点，则边子集$F\subseteq E(G)$称为限制边割。$G$的限制边连通$λ'(G)$是$G$的最小限制边割的基数，如果它有任何;否则是$λ'(G)=+\infty$。如果$G$不是星型，且其阶数至少为4，则为$λ'(G)\leq ξ(G)$，其中$ξ(G)=$ min $\{d_G(u) + d_G(v)-2:\ uv \in E(G)\}$。图$G$被称为最大限制边连通的，如果$λ'(G)= ξ(G)$ ;图$G$被称为超限制边连通的，如果每个最小限制边割从$G$中隔离出一条边。图$G_1$和$G_2$的直积，记作$G_1\times G_2$，是顶点集为$V(G_1\times G_2) = V(G_1)\times V(G_2)$的图，其中两个顶点$(u_{1} ,v_{1} )$和$(u_{2} ,v_{2} )$在$G_1\times G_2$中相邻当且仅当$u_{1}u_{2} \in E(G_1)$和$v_{1}v_{2} \in E(G_2)$ .本文给出了$G\times K_{n}$是超限制边连通的一个充分条件，其中$K_{n}$是$n$顶点上的完全图。
英文摘要：
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. An edge subset $F\subseteq E(G)$ is called a restricted edge-cut if $G-F$ is disconnected and has no isolated vertices. The restricted edge-connectivity $λ'(G)$ of $G$ is the cardinality of a minimum restricted edge-cut of $G$ if it has any; otherwise $λ'(G)=+\infty$. If $G$ is not a star and its order is at least four, then $λ'(G)\leq ξ(G)$, where $ξ(G)=$ min$\{d_G(u) + d_G(v)-2:\ uv \in E(G)\}$. The graph $G$ is said to be maximally restricted edge-connected if $λ'(G)= ξ(G)$; the graph $G$ is said to be super restricted edge-connected if every minimum restricted edge-cut isolates an edge from $G$. The direct product of graphs $G_1$ and $G_2$, denoted by $G_1\times G_2$, is the graph with vertex set $V(G_1\times G_2) = V(G_1)\times V(G_2)$, where two vertices $(u_{1} ,v_{1} )$ and $(u_{2} ,v_{2} )$ are adjacent in $G_1\times G_2$ if and only if $u_{1}u_{2} \in E(G_1)$ and $v_{1}v_{2} \in E(G_2)$. In this paper, we give a sufficient condition for $G\times K_{n}$ to be super restricted edge-connected, where $K_{n}$ is the complete graph on $n$ vertices.
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。
