Orthogonal embeddings of graphs in Euclidean space

Abstract

Let G = ( V , E ) be a simple connected graph. An injective function f : V → R n is called an n -dimensional (or n -D) orthogonal labeling of G if u v , u w ∈ 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 R n orthogonally. Let the orthogonal rank o r ( G ) of G be the minimum value of n , where G admits an n -D orthogonal labeling (otherwise, we define o r ( G ) = ∞ ). 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.