Small Planar Hypohamiltonian Graphs

Abstract

A graph is hypohamiltonian if it is non-hamiltonian, but the deletion of every single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 40 vertices, a result due to Jooyandeh, McKay, Östergård, Pettersson, and Zamfirescu. That result is here improved upon by two planar hypohamiltonian graphs on 34 vertices. We exploited a special subgraph contained in two graphs of Jooyandeh et al., and modified it to construct the two 34-vertex graphs and six planar hypohamiltonian graphs on 37 vertices. Each of the 34-vertex graphs has 26 cubic vertices, improving upon the result of Jooyandeh et al. that planar hypohamiltonian graphs have 30 cubic vertices. We use the 34-vertex graphs to construct hypohamiltonian graphs of order 34 with crossing number 1, improving the best-known bound of 36 due to Wiener. Whether there exists a planar hypohamiltonian graph on 41 vertices was an open question. We settled this question by applying an operation introduced by Thomassen to the 37-vertex graphs to obtain several planar hypohamiltonian graphs on 41 vertices. The 25 planar hypohamiltonian graphs on 40 vertices of Jooyandeh et al. have no nontrivial automorphisms. The result is here improved upon by six planar hypohamiltonian graphs on 40 vertices with nontrivial automorphisms.