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Structural and computational results on platypus graphs. (English) Zbl 1462.05110

Summary: A platypus graph is a non-hamiltonian graph for which every vertex-deleted subgraph is traceable. They are closely related to families of graphs satisfying interesting conditions regarding longest paths and longest cycles, for instance hypohamiltonian, leaf-stable, and maximally non-hamiltonian graphs.
In this paper, we first investigate cubic platypus graphs, covering all orders for which such graphs exist: in the general and polyhedral case as well as for snarks. We then present (not necessarily cubic) platypus graphs of girth up to 16 – whereas no hypohamiltonian graphs of girth greater than 7 are known – and study their maximum degree, generalising two theorems of G. Chartrand et al. [in: Second international conference on combinatorial mathematics, New York, 1978. New York, NY: The New York Academy of Sciences. 130–135 (1979; Zbl 0481.05039)]. Using computational methods, we determine the complete list of all non-isomorphic platypus graphs for various orders and girths. Finally, we address two questions raised by the third author in [J. Graph Theory 86, No. 2, 223–243 (2017; Zbl 1370.05115)].

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C38 Paths and cycles
05C45 Eulerian and Hamiltonian graphs
05C85 Graph algorithms (graph-theoretic aspects)
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References:

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