Dual Circumference and Collinear Sets

Abstract

We show that, if an n-vertex triangulation G of maximum degree \(\Delta \) has a dual that contains a cycle of length \(\ell \), then G has a non-crossing straight-line drawing in which some set, called a collinear set, of \(\Omega (\ell /\Delta ^4)\) vertices lie on a line. Using the current lower bounds on the length of longest cycles in cubic 3-connected graphs, this implies that every n-vertex planar graph of maximum degree \(\Delta \) has a collinear set of size \(\Omega (n^{0.8}/\Delta ^4)\).