Full friendly index sets of Cartesian products of two cycles

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 υ _f (i) = |f ⁻¹(i)| and e _f (i) = |f*⁻¹(i)|. A labeling f is called friendly if |υ _f (1) − υ _f (0)| ≤ 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 will determine the full friendly index set of every Cartesian product of two cycles.