Full friendly index sets of cylinder graphs

Let G = (V,E) be a connected simple graph. A labeling f : V → Z2 induces an edge labeling f : E → Z2 defined by f(xy) = f(x) + f(y) for each xy ∈ E. For i ∈ Z2, let vf (i) = |f−1(i)| and ef(i) = |(f+)−1(i)|. A labeling f is called friendly if |vf (1) − vf(0)| ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if(G) = ef (1) − ef(0). The set {if(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 Cm × Pn for even m ≥ 4, even n ≥ 4 and m ≤ 2n. We also list the results of other cases for m,n ≥ 4.