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 \(\Omega (n^{0.8}/\Delta ^4)\).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
