vector labelling
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10.37236/703 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Ross J. Kang ◽  
László Lovász ◽  
Tobias Müller ◽  
Edward R. Scheinerman

A graph $G$ is a $k$-dot product graph if there exists a vector labelling $u: V(G) \to \mathbb{R}^k$ such that $u(i)^{T}u(j) \geq 1$ if and only if $ij \in E(G)$. Fiduccia, Scheinerman, Trenk and Zito [Discrete Math., 1998] asked whether every planar graph is a $3$-dot product graph. We show that the answer is "no". On the other hand, every planar graph is a $4$-dot product graph. We also answer the corresponding questions for planar graphs of prescribed girth and for outerplanar graphs.


1998 ◽  
Vol 43 (5) ◽  
pp. 381-383
Author(s):  
Jianghua Fan ◽  
Zeke Wang

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