## Abstract

Let G = (V,E) be a simple connected graph. An injective function f: V R^{n} is called an ndimensional (or n-D) orthogonal labeling of G if uv, uw n E implies that (f(v)-f(u)) (f(w)- f(u)) = 0, where is the usual dot product in Euclidean space. If such an orthogonal labeling f of G exists, then G is said to be embedded in Rn orthogonally. Let the orthogonal rank or(G) of G be the minimum value of n, where G admits an n-D orthogonal labeling (otherwise, we define or(G) = 8). In this paper, we establish some general results for orthogonal embeddings of graphs. We also determine the orthogonal ranks for cycles, complete bipartite graphs, one-point union of two graphs, Cartesian product of orthogonal graphs, bicyclic graphs without pendant, and tessellation graphs.

Original language | English |
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Pages (from-to) | 373-381 |

Number of pages | 9 |

Journal | Electronic Journal of Graph Theory and Applications |

Volume | 7 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2019 |

## Scopus Subject Areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

## User-Defined Keywords

- Euclidean space 2010 Mathematics Subject Classification
- Graph embedding
- Orthogonal drawing
- Orthogonal labeling