Cubic graphs have bounded slope parameter

We show that every finite connected graph G with maximum degree three and with at least one vertex of degree smaller than three has a straight-line drawing in the plane satisfying the following conditions. No three vertices are collinear, and a pair of vertices form an edge in G if and only if the segment connecting them is parallel to one of the sides of a previously fixed regular pentagon. It is also proved that every finite graph with maximum degree three permits a straight-line drawing with the above properties using at most seven different edge slopes. Submitted: October 2008 Reviewed: July 2009 Revised: August 2009 Accepted: November 2009 Final: November 2009 Published: January 2010 Article type: Regular paper Communicated by: I. G. Tollis and M. Patrignani Research supported by NSF Grant CCF-05-14079, and by grants from NSA, PSC-CUNY, BSF and the Hungarian Research Foundation OTKA E-mail addresses: [email protected] (Balázs Keszegh) [email protected] (János Pach) [email protected] (Dömötör Pálvölgyi) [email protected] (Géza Tóth) 6 Keszegh et al. Cubic graphs have bounded slope parameter