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Non-Hamiltonian graphs in which every edge-contracted subgraph is Hamiltonian. (English) Zbl 1462.05206

Summary: A graph \(G\) is perihamiltonian if \(G\) itself is non-hamiltonian, yet every edge-contracted subgraph of \(G\) is hamiltonian. These graphs form a superclass of the hypohamiltonian graphs. By applying a recent result of Wiener on path-critical graphs, we prove the existence of infinitely many perihamiltonian graphs of connectivity \(k\) for any \(k \geq 2\). We also show that every planar perihamiltonian graph of connectivity \(k\) contains a vertex of degree \(k\). This strengthens a theorem of Thomassen, and entails that if in a polyhedral graph of minimum degree at least 4 the set of vertices whose removal yields a non-hamiltonian graph is independent, the graph itself must be hamiltonian. Finally, while we here prove that there are infinitely many polyhedral perihamiltonian graphs containing no adjacent cubic vertices, whether an analogous result holds for the hypohamiltonian case remains open.

MSC:

05C45 Eulerian and Hamiltonian graphs
05C10 Planar graphs; geometric and topological aspects of graph theory
05C40 Connectivity
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