Paper Info
Title
Cubic vertices in planar hypohamiltonian graphs.
Abstract
Thomassen showed in 1978 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 Brinkmann and the author, 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 J. Graph Theory [79 (2015) 63-81]), 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.
Year
DOI
Venue
2019
10.1002/jgt.22388
JOURNAL OF GRAPH THEORY
Keywords
DocType
Volume
hypohamiltonian,planar,3-connected,3-cut
Journal
90.0
Issue
ISSN
Citations 
2.0
0364-9024
0
PageRank 
References 
Authors
0.34
0
1
Name
Order
Citations
PageRank
Carol T. Zamfirescu13815.25