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Wai Chee Shiu
Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P.R. China
Electrical & Electronics Engineering

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Orthogonal embeddings of graphs in Euclidean space Wai Chee Shiu; Richard M. Low
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 7, No 2 (2019): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2019.7.2.13

Abstract

Let G = (V, E) be a simple connected graph. An injective function f : V → Rn is called an n-dimensional (or n-D) orthogonal labeling of G if uv, uw ∈ 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) = ∞). 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.
Co-Authors Richard M. Low