Fractional Domination of the Cartesian Products in Graphs

Let G = (V,E) be a simple graph. For any real function g : V −→ R and a subset S ⊆ V , we write g(S) = ∑ v∈S g(v). A function f : V −→ [0, 1] is said to be a fractional dominating function (FDF ) of G if f(N [v]) ≥ 1 holds for every vertex v ∈ V (G). The fractional domination number γf (G) of G is defined as γf (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)) ≥ 1 instead of f(N [v]) ≥ 1. The fractional total domination number γ f (G) of G is analogous. In this note we give the exact values of γf (Cm × Pn) and γ f (Cm × Pn) for all integers m ≥ 3 and n ≥ 2.