The friendly index set of Pm × Pn

Abstract

For a graph G = (V, E) and a binary labeling (coloring) f : V(G) → Z₂, let v_f(i) = |f^-1(i)| f is said to be friendly if |vf(1) - v_f(0)| ≤ 1. The labeling f : V(G) → Z₂ induces an edge labeling f* : E(G) → Z₂ defined by f*(xy) = |f(x) - f(y)| ∀ xy € E(G). Let e _f,(i) = |f*^-1(i)|. The friendly index set of the graph G, denoted by FI(G), is defined by FI(G) = {|e_f(1) - e _f(0)| : f is a friendly vertx labeling of G }. In this paper we determine the friendly index set of P_m × P_n.