Full friendly index sets of cylinder graphs

Let G = (V,E) be a connected simple graph. A labeling f: V → ℤ ₂induces an edge labeling f ⁺: E → ℤ ₂ defined by f ⁺(xy) = f(x) + f(y) for each xy ∈ E. For i ∈ ℤ ₂, let v _f(i) = {pipe}f ^-1(i){pipe} and e _f(i) = {pipe}(f ⁺) ^-1(i){pipe}. A labeling f is called friendly if {pipe}v _f(1)-v _f(0){pipe} ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by i _f(G)=e _f(1)-e _f(0). The set {i _f(G)f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI(G). In this paper, we determine the full friendly index sets of cylinder graphs C _m × P _n for even m ≥ 4, even n ≥ 4 and m ≤ 2n. We also list the results of other cases for m, n ≥ 4.