2 research outputs found
Orthogonal embeddings of graphs in Euclidean space
Let Gβ=β(V,βE) be a simple connected graph. An injective function fβ:βVβββRn is called an n-dimensional (or n-D) orthogonal labeling of G if uv,βuwβββE implies that (f(v)β
ββ
f(u))β
β
β
(f(w)β
ββ
f(u))β=β0, where β
β
β
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)β=ββ). 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.</p