## Abstract

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 P_{2} 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).

Original language | English (US) |
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Pages (from-to) | 221-228 |

Number of pages | 8 |

Journal | AKCE International Journal of Graphs and Combinatorics |

Volume | 10 |

Issue number | 2 |

State | Published - Aug 1 2013 |

## Keywords

- Dumbbell
- Generalized prism
- Odd harmonious graphs
- Odd harmonious labeling