Fractional Domination of the Cartesian Products in Graphs

Abstract:

      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 $n\geq 2$.