Fractional Domination of the Cartesian Products in Graphs
Received:March 01, 2014  Revised:January 16, 2015
Key Words: Cartesian products   fractional domination number   fractional total domination number  
Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11361024; 11061014) and the Jiangxi Provincial Science and Technology Project (Grant No.KJLD12067).
Author NameAffiliation
Baogen XU Department of Mathematics, East China Jiaotong University, Jiangxi 330013, P. R. China 
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      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 $n\geq 2$.
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