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Regular non-Hamiltonian polyhedral graphs. (English) Zbl 1427.05126
Summary: Invoking Steinitz’ theorem, in the following a polyhedron shall be a 3-connected planar graph. From around 1880 till 1946 Tait’s conjecture that cubic polyhedra are Hamiltonian was thought to hold – its truth would have implied the Four Colour theorem. However, Tutte gave a counterexample. We briefly survey the ensuing hunt for the smallest non-Hamiltonian cubic polyhedron, the Lederberg-Bosák-Barnette graph, and prove that there exists a non-Hamiltonian essentially 4-connected cubic polyhedron of order \(n\) if and only if \(n\geq 42\). This extends work of R. E. L. Aldred et al. [SIAM J. Discrete Math. 13, No. 1, 25–32 (2000; Zbl 0941.05041)]. We then present our main results which revolve around the quartic case: combining a novel theoretical approach for determining non-hamiltonicity in (not necessarily planar) graphs of connectivity 3 with computational methods, we dramatically improve two bounds due to J. Zaks [J. Comb. Theory, Ser. B 21, 116–131 (1976; Zbl 0309.05120)]. In particular, we show that the smallest non-Hamiltonian quartic polyhedron has at least 35 and at most 39 vertices, thereby almost reaching a quartic analogue of a famous result of D. A. Holton and B. D. McKay [ibid. 45, No. 3, 305–319 (1988; Zbl 0607.05051)]. As an application of our results, we obtain that the shortness coefficient of the family of all quartic polyhedra does not exceed 5/6. The paper ends with a discussion of the quintic case in which we tighten a result of Owens.

05C45 Eulerian and Hamiltonian graphs
05C10 Planar graphs; geometric and topological aspects of graph theory
05C38 Paths and cycles
05C35 Extremal problems in graph theory
Full Text: DOI
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