## Abstract

Let G be a graph of order n and size e. A vertex-magic total labeling is an assignment of the integers 1, 2, ..., n + e to the vertices and the edges of G, so that at each vertex, the vertex label and the labels on the edges incident at that vertex, add to a fixed constant, called the magic number of G. Such a labeling is a-vertex consecutive magic if the set of the labels of the vertices is { a + 1, a + 2, ..., a + n }, and is b-edge consecutive magic if the set of labels of the edges is { b + 1, b + 2, ..., b + e }. In this paper we prove that if an a-vertex consecutive magic graph has isolated vertices then the order and the size satisfy (n - 1)^{2} + n^{2} = (2 e + 1)^{2}. Moreover, we show that every tree with even order is not a-vertex consecutive magic and, if a tree of odd order is a-vertex consecutive then a = n - 1. Furthermore, we show that every a-vertex consecutive magic graph has minimum degree at least two if a = 0, or both 2 e ≥ sqrt(6 n^{2} - 2 n + 1) and 2 a ≤ e, and the minimum degree is at least three if both 2 e ≥ sqrt(10 n^{2} - 6 n + 1 + 4 a) and 2 a ≤ e. Finally, we state analogous results for b-edge consecutive magic graphs.

Original language | English |
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Pages (from-to) | 1817-1829 |

Number of pages | 13 |

Journal | Discrete Mathematics |

Volume | 306 |

Issue number | 16 |

DOIs | |

Publication status | Published - 28 Aug 2006 |

## Keywords

- Consecutive magic labeling
- Super vertex-magic labeling
- Vertex-magic labeling