## Abstract

A graph G with q edges is said to be harmonious if there is an injection f from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(. x). +. f(. y). (mod. q), the resulting edge labels are distinct. If G is a tree, exactly one label may be used on two vertices. Over the years, many variations of harmonious labelings have been introduced.We study a variant of harmonious labeling. A function f is said to be a properly even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2(q-1) and the induced function f ^{*} from the edges of G to 0, 2, . . ., 2(q-1) defined by f ^{*} (xy)=f(x)+f(y)(mod2q) is bijective. We investigate the existence of properly even harmonious labelings of families of disconnected graphs with one of C _{3} , C _{4} , K _{4} or W _{4} as a component.

Original language | English (US) |
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Pages (from-to) | 204-215 |

Number of pages | 12 |

Journal | AKCE International Journal of Graphs and Combinatorics |

Volume | 12 |

Issue number | 2-3 |

DOIs | |

State | Published - Nov 1 2015 |

### Bibliographical note

Publisher Copyright:© 2015 Kalasalingam University.

## Keywords

- Even harmonious labelings
- Graph labelings
- Harmonious labelings
- Properly even harmonious labelings