Regular non-hamiltonian polyhedral graphs

Nico Van Cleemput, Carol T. Zamfirescu
<span title="">2018</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/xkk23xlywfez5l246cnmlit4qq" style="color: black;">Applied Mathematics and Computation</a> </i> &nbsp;
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 nonhamiltonian cubic polyhedron, the Lederberg-Bosák-Barnette graph, and prove that there exists a non-hamiltonian essentially 4-connected cubic polyhedron of
more &raquo; ... order n if and only if n ≥ 42. This extends work of Aldred, Bau, Holton, and McKay. 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 Zaks. 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 Holton and McKay. 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. 193 polyhedron contains at least one 3-cut. For more on the interplay between polyhedra, cuts, and hamiltonicity, we refer to the survey [32] . Let G be a polyhedron of connectivity 3 and X The cubic case Tait conjectured [35] in 1884 that every cubic polyhedron is hamiltonian. This conjecture became famous because it implied the Four Colour Theorem (which at that time was itself open): by Jordan's Curve Theorem, any hamiltonian cycle h in a cubic polyhedron G naturally divides the plane into an unbounded region A and a bounded region B with A ∩ B = h . The duals of the planar graphs G ∩ A and G ∩ B are trees, so we may colour their vertices alternatingly. Thus, we can colour the faces of G with four colours such that no two adjacent faces receive the same colour. One can reduce the case of general polyhedra to cubic polyhedra, and thus, Tait's conjecture would have implied the Four Colour Theorem. However, Tait's conjecture turned out to be false and the first to construct a counterexample was Tutte in 1946, see [38] , using in his approach three copies of a graph that has become to be known as "Tutte-fragment". The smallest counterexample is due to Lederberg (and independently, Bosák and Barnette), has order 38, and is also based on Tutte-fragments. ( To be precise, there are six structurally very similar such graphs [19] .) That this is indeed the smallest possible counterexample was shown by Holton and McKay [19] after a long series of papers by various authors, see for instance work of Butler [10] , Barnette and Wegner [2] , and Okamura [28] . The second part of the next theorem follows directly from the Holton-McKay result by successively substituting a vertex with a triangle. Theorem 1 (Holton and McKay [19] ) . We have that Furthermore, for each even n ≥ 38 there is a non-hamiltonian cubic polyhedron on n vertices. Balinski asked whether non-traceable cubic polyhedra exist. Brown and independently Grünbaum and Motzkin proved the existence of such graphs. Klee asks for determining p 3 . (We refer to Klee's excellent [13, Chapter 17] for references and further details.) The best bounds that are known are as follows. Theorem 2 (Knorr [23] and T. Zamfirescu [44] ) . We have that
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