Lecture Notes in Computer Science Volume 5417, 2009, pp 50-60

Cubic Graphs Have Bounded Slope Parameter

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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 only at most seven different edge slopes.