The Odd Harmonious Labeling of Dumbbell and Generalized Prism Graphs

Abstract

A graph G = (V, E) with |E| = q is said to be odd harmonious if there exists an injection f: V (G) → {0, 1, 2,…, 2q - 1} such that the induced function f*: E(G) → {1, 3, 5,…, 2q - 1} defined by f*(xy) = f(x)+f(y) is a bijection. Then f is said to be odd harmonious labeling of G. A dumbbell graph Dn,k,2 is a bicyclic graph consisting of two vertex-disjoint cycles Cn, Ck and a path P2 joining one vertex of Cn with one vertex of Ck. A prism graph Cn x Pm is a Cartesian product of cycle Cn and path Pm. In this paper we show that the dumbbell graph Dn,k,2 is odd harmonious for n ≡ k ≡ 0 (mod 4) and n ≡ k ≡ 2 (mod 4), generalized prism graph Cn x Pm is odd harmonious for n ≡ 0 (mod 4) and for any m, and generalized prism graph Cn x Pm is not odd harmonious for n ≡ 2 (mod 4).