Cubic vertices in planar hypohamiltonian graphs. (English) Zbl 1466.05038

Summary: C. Thomassen [Discrete Math. 14, 377–389 (1976; Zbl 0322.05130)] showed that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with minimum degree at least 4 in which every vertex-deleted subgraph is hamiltonian, must be itself hamiltonian. By applying work of G. Brinkmann and C. T. Zamfirescu [Electron. J. Comb. 26, No. 1, Research Paper P1.39, 16 p. (2019; Zbl 1409.05122)], we extend this result in three directions. We prove that (i) every planar hypohamiltonian graph contains at least four cubic vertices, (ii) every planar almost hypohamiltonian graph contains a cubic vertex, which is not the exceptional vertex (solving a problem of the author raised in [C. T. Zamfirescu, J. Graph Theory 79, No. 1, 63–81 (2015; Zbl 1312.05076)], and (iii) every hypohamiltonian graph with crossing number 1 contains a cubic vertex. Furthermore, we settle a recent question of Thomassen by proving that asymptotically the ratio of the minimum number of cubic vertices to the order of a planar hypohamiltonian graph vanishes.


05C07 Vertex degrees
05C10 Planar graphs; geometric and topological aspects of graph theory
05C45 Eulerian and Hamiltonian graphs
Full Text: DOI


This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.