## Abstract

A graph G = (V, E) with p vertices and q edges is called edge-magic if there is a bijection f : E → {1, 2, ..., q} such that the induced mapping f^{+} : V → Z_{p} is a constant mapping, where f^{+} (u) ≡ ∑ uv ∈ E f(uv) (mod p). A necessary condition of edge-magicness is p {divides} q(q+1). The edge magic index of a graph G is the least positive integer k such that the k-fold of G is edge-magic. In this paper, we prove that for any multigraph G with n vertices, n - 1 edges having no loops and no isolated vertices, the k-fold of G is edge-magic if n and k satisfy a necessary condition for edge-magicness and n is odd. For n even we also have some results on full m-ary trees and spider graphs. Some counterexamples of the edge-magic indices of trees conjecture are given.

Original language | English |
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Pages (from-to) | 443-458 |

Number of pages | 16 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 11 |

DOIs | |

Publication status | Published - Jul 2002 |

## Scopus Subject Areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

## User-Defined Keywords

- Edge-magic
- edge-magic index
- spider graph
- tree