On Friendly Index Sets of Broken Wheels with Three Spokes

摘要

Let G be a graph with vertex set V(G) and edge set E(G), and let A be an abelian group. A labeling f : V(G) → A induces a edge labeling f~* : E(G) → A defined by f~*(xy) = f(x) + f(y) for each xy ∈ E. For each i ∈ A, let v_f(i) = card{v ∈V(G) : f(v) = i} and e_f(i) = card{e ∈ E(G) : f~*(e) = i}. Let c(f) = {|e_f(i) - e_f(j)| : (I,j) ∈ A × A}. A labeling f of a graph G is said to be A-friendly if |v_f(i) - v_f(j)| ≤ 1 for all (I,j) ∈ A × A. If c(f) is a (0, 1)-matrix for an A-friendly labeling f, then f is said to be A-cordial. When A = Z2, the friendly index set of the graph G, FI(G), is defined as {|e_f(0) - e_f(1)| : the vertex labeling f is Z_2-friendly}. In [15] the friendly index set of a cycle is completely determined. We consider the friendly index sets of broken wheels with three spokes.