Full friendly index sets of cylinder graphs
Wai Chee Shiu, Fook Sun Wong · 2012
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. Fori∈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.