\documentclass[12pt]{article} \usepackage{amsmath,mathrsfs,bbm} \usepackage{amssymb} \textwidth=4.825in \overfullrule=0pt \thispagestyle{empty} \begin{document} \noindent % % {\bf Makoto Araya and G\'abor Wiener } % % \medskip \noindent % % {\bf On Cubic Planar Hypohamiltonian and Hypotraceable Graphs} % % \vskip 5mm \noindent % % % % We present a cubic planar hypohamiltonian graph on 70 vertices, improving the best known bound of 94 by Thomassen and derive some consequences concerning longest paths and cycles of planar $3$-connected graphs. We also show that cubic planar hypohamiltonian graphs on $n$ vertices exist for every even number $n\geq 86$ and that cubic planar hypotraceable graphs exist on $n$ vertices for every even number $n \geq 356$, settling an open question of Holton and Sheehan. \end{document} .