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. Copyright © 2010 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg