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On planar hypohamiltonian graphs. (English) Zbl 1223.05168

Summary: We present a planar hypohamiltonian graph on 42 vertices and (as a corollary) a planar hypotraceable graph on 162 vertices, improving the bounds of Zamfirescu and Zamfirescu and show some other consequences. We also settle the open problem whether there exists a positive integer \(N\), such that for every integer \(n\geq N\) there exists a planar hypohamiltonian/hypotraceable graph on \(n\) vertices.

MSC:

05C45 Eulerian and Hamiltonian graphs
05C10 Planar graphs; geometric and topological aspects of graph theory
05C38 Paths and cycles
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