Odd Vertex Equitable Even Labeling of Cycle Related Graphs

Abstract

Let G be a graph with p vertices and q edges and A = {1, 3, ..., q} if q is odd or A = {1, 3, ..., q + 1} if q is even. A graph G is said to admit an odd vertex equitable even labeling if there exists a vertex labeling f : V(G) → A that induces an edge labeling f^∗ defined by f^∗ (uv) = f(u) + f(v) for all edges uv such that for all a and b in A, |v_f(a) − v_f(b)| ≤ 1 and the induced edge labels are 2, 4, ..., 2q where v_f(a) be the number of vertices v with f(v) = a for a ∈ A. A graph that admits an odd vertex equitable even labeling is called an odd vertex equitable even graph. Here, we prove that the subdivision of double triangular snake (S(D(T_n))), subdivision of double quadrilateral snake (S(D(Q_n))), DA(Q_m) ⊙ nK₁ and DA(T_m) ⊙ nK₁ are odd vertex equitable even graphs.