Fractional Domination of the Cartesian Products in Graphs

DOI：10.3770/j.issn:2095-2651.2015.03.005

 作者 单位 徐保根 东交通大学数学系, 江西 南昌 330013

设$G=(V,E)$为一个简单图,对于任意实函数 $g：V\longrightarrow R$和一个子集 $S\subseteq V$,则记~$g(S)=\sum_{v \in S}g(v)$.一个函数 $f:V\longrightarrow [0,1]$ 如果满足$f(N[v])\geq 1$对每个点$v\in V(G)$成立,则称$f$为图$G$的一个Fractional控制函数,且图$G$的Fractional控制数定义为$\gamma_{f}(G)=\min\{f(V)|f$为图$G$的Fractional控制函数\}.图$G$的Fractional全控制函数可类似地定义,其区别是将$f(N(v))\geq 1$代替$f(N[v])\geq 1$.图$G$的Fractional全控制数$\gamma_{f}^{0}(G)$是类似的.在本文中,对所有的整数$m\geq 3$和$n\geq 2$,给出了$\gamma_{f}(C_{m}\times P_{n})$ 和$\gamma_{f}^{0}(C_{m}\times P_{n})$的确切值.

Let $G=(V,E)$ be a simple graph. For any real function $g：V\longrightarrow R$ and a subset $S\subseteq V$, we write $g(S)=\sum_{v \in S}g(v)$. A function $f:V\longrightarrow [0,1]$ is said to be a fractional dominating function $(FDF)$ of $G$ if $f(N[v])\geq 1$ holds for every vertex $v\in V(G)$. The fractional domination number $\gamma_{f}(G)$ of $G$ is defined as $\gamma_{f}(G)=\min \{f(V)|f$ is an $FDF$ of $G$ \}. The fractional total dominating function $f$ is defined just as the fractional dominating function, the difference being that $f(N(v))\geq 1$ instead of $f(N[v])\geq 1$. The fractional total domination number $\gamma_{f}^{0}(G)$ of $G$ is analogous. In this note we give the exact values of $\gamma_{f}(C_{m}\times P_{n})$ and $\gamma_{f}^{0}(C_{m}\times P_{n})$ for all integers $m\geq 3$ and $n\geq 2$.