Region indices of friendly labeling of a planar graph

Abstract

Let G = (V, E) be a planar graph embedded on a plane or plane graph. If R the set of all regions r, V' be the set of cut vertices in G, V_τ the set of vertices in region r and b_v is the number of connected components in the graph G - v then vertex labeling f: V → ℤ₂ induces a region labeling f^∗: R → ℤ₂ which is defined as: (Equation presented)r is a region bounded by a cycle. (mod 2) r is a region not bounded by a cycle. For each, i ∈ ℤ₂ define u_f(i) = |f-¹(i)| and r_f(i) = |f^∗-1(i)|.Wq Call f friendly if, |u_f(1) - v_f(0)| ≤ 1. The full region index set of friendly labeling of G, FRIFL(G) is defined as {r_f(1) - r_f(0): f is friendly labeling}. In this paper, we study the full region index set of friendly labeling of cycle C_n, wheel W_n, fan F_m, cartesian product of path P₂ and P_n i.e. P₂ × P_n.