2013/02/12 by Mohammadreza Jooyandeh, Brendan D. McKay, Jooyandeh, Mohammadreza +7 · 1 citation
Computer Science · #05C10 #05C30 #05C38 #05C45 #05C85 #Advanced Graph Theory Research #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Graph Labeling and Dimension Problems
paper · pdf · doi:10.48550/arxiv.1302.2698
openalex publication_date 2013/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer-aided generation of certain families of planar graphs with girth 4 and a fixed number of 4-faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamiltonian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest hypohamiltonian planar graph of girth 5 has 45 vertices.