The odd harmonious labeling of dumbbell and generalized prism graphs

A graph G = (V, E) with {pipe}E{pipe} = q is said to be odd harmonious if there exists an injection f : V(G) → {0, 1, 2,..., 2_q -1} such that the induced function f*: E(G) → {1, 3, 5,..., 2_q -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 D_n,k,2 is a bicyclic graph consisting of two vertex-disjoint cycles C_n, C_k and a path P₂ joining one vertex of C_n with one vertex of C_k. A prism graph C_n × P_m is a Cartesian product of cycle C_n and path P_m. In this paper we show that the dumbbell graph D_n,k,2 is odd harmonious for n ≡ k ≡ 0 (mod 4) and n ≡ k ≡ 2 (mod 4), generalized prism graph C_n × P_m is odd harmonious for n ≡ 0 (mod 4) and for any m, and generalized prism graph C_n × P_m is not odd harmonious for n ≡ 2 (mod 4).