Extreme Friendly Indices of C_m × P_n

Let G = (V,E) be a connected simple graph. A labeling ƒ: V → Z₂ induces an edge labeling ƒ* : E → Z₂ defined by ƒ*(xy) = ƒ(x) + ƒ(y) for each xy ∈ E. For i ∈ Z₂, let v_ƒ(i) = |ƒ^-1(i)| and e_ƒ(i) = |ƒ*^-1(i)|. If |v_ƒ(1)-v_ƒ(0)| ≤ 1, then ƒ is called a friendly labeling of G. For a friendly labeling f of a graph G, we define the friendly index of G under ƒ by i_ƒ(G) = e_ƒ(1) - e_ƒ(0). The set {i_ƒ(G)|ƒ is a friendly labeling of G} is called the full friendly index set of G. In this paper, we will present the extreme friendly indices, i.e., the maximum and minimum friendly indices of Cartesian product of a cycle and a path.