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On hypohamiltonian and almost hypohamiltonian graphs. (English) Zbl 1312.05076

Summary: A graph \(G\) is almost hypohamiltonian if \(G\) is non-Hamiltonian, there exists a vertex \(w\) such that \(G-v\) is non-Hamiltonian, and for any vertex \(v\neq w\) the graph \(G-v\) is Hamiltonian. We prove the existence of an almost hypohamiltonian graph with 17 vertices and of a planar such graph with 39 vertices. Moreover, we find a 4-connected almost hypohamiltonian graph, while Thomassen’s question whether 4-connected hypohamiltonian graphs exist remains open. We construct planar almost hypohamiltonian graphs of order \(n\) for every \(n\geq 74\). During our investigation we draw connections between hypotraceable, hypohamiltonian, and almost hypohamiltonian graphs, and discuss a natural extension of almost hypohamiltonicity. Finally, we give a short argument disproving a conjecture of Chvátal (originally disproved by Thomassen), strengthen a result of Araya and Wiener on cubic planar hypohamiltonian graphs, and mention open problems.

MSC:

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