## Abstract

Let G = (V, E) be a graph of order n. A distance magic labeling of G is a bijection ℓ: V → {1, 2, . . ., n} for which there exists a positive integer k such that ^{P} _{x} ∈N(v) ℓ(x) = k for all v ∈ V, where N(v) is the open neighborhood of v. Tuttes flow conjectures are a major source of inspiration in graph theory. In this paper we ask when we can assign n distinct labels from the set {1, 2, . . ., n} to the vertices of a graph G of order n such that the sum of the labels on heads minus the sum of the labels on tails is constant modulo n for each vertex of G. Therefore we generalize the notion of distance magic labeling for oriented graphs.

Original language | English (US) |
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Pages (from-to) | 533-546 |

Number of pages | 14 |

Journal | Discussiones Mathematicae - Graph Theory |

Volume | 39 |

Issue number | 1 |

DOIs | |

State | Published - 2019 |

### Bibliographical note

Funding Information:1This work was partially supported by the Faculty of Applied Mathematics AGH UST statutory tasks within subsidy of Ministry of Science and Higher Education.

## Keywords

- Digraph
- Distance magic graph
- Flow graph