Hamiltonian cycles in cubic 3-connected bipartite planar graphs

Holton, Derek; Manvel, Bennet; McKay, Brendan D.

doi:10.1016/0095-8956(85)90072-3

Cited by 58 publications

(32 citation statements)

References 12 publications

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“…Goodey [77] obtained significant results on this problem, and an excellent account is found in Fleischner's book [69]. Other partial results on Barnette's conjecture have been obtained by Fleischner [65], [68], by Fouquet and Thuillier [72], by Peterson [121], and by Holton, Manvel, and McKay [84], and by Plummer and Pulleyblank [122]. Related conjectures are discussed in [70] and [72].…”

Section: Sufficient Conditionsmentioning

confidence: 78%

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Supereulerian graphs: A survey

Catlin

1992

Journal of Graph Theory

136

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Section: Sufficient Conditionsmentioning

confidence: 78%

“…Fleischner [68] reviewed work on cycle decompositions, and he has discussed problems involving decompositions of and transitions in eulerian graphs [69]. For more references concerning hamiltonian 3-regular graphs, see [84]. Research on snarks (graphs not in S 3 that are minimal in a certain sense) is discussed in [60] and [71].…”

Section: Other Remarksmentioning

confidence: 99%

Supereulerian graphs: A survey

Catlin

1992

Journal of Graph Theory

136

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“…This result was motivated by our observation that the sequence of numbers of main classes of spherical Latin bitrades (determined by Wanless [9] as a by-product of his exhaustive classification of Latin bitrades of size up to 19) is equal, up to a shift, to the sequence of numbers of non-isomorphic planar Eulerian triangulations obtained by Holton, Manvel and McKay [5]. (For completeness let us note that the article [5] deals with the enumeration of the dual objects, that is, cubic 3-connected bipartite planar graphs.…”

Section: The Bijectionmentioning

confidence: 94%

Planar Eulerian triangulations are equivalent to spherical Latin bitrades

Cavenagh

Lisonĕk

2008

Journal of Combinatorial Theory, Series A

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Given a pair of Latin squares, we may remove from both squares those cells that contain the same symbol in corresponding positions. The resulting pair T = {P 1 , P 2 } of partial Latin squares is called a Latin bitrade. The number of filled cells in P 1 is called the size of T . There are at least two natural ways to define the genus of a Latin bitrade; the bitrades of genus 0 are called spherical. We construct a simple bijection between the isomorphism classes of planar Eulerian triangulations on v vertices and the main classes of spherical Latin bitrades of size v − 2. Since there exists a fast algorithm (due to Batagelj, Brinkmann and McKay) for generating planar Eulerian triangulations up to isomorphism, our result implies that also spherical Latin bitrades can be generated very efficiently.A partial Steiner triple system is a set of 3-subsets ("triples") of a finite ground set V such that each 2-set of points occurs as a subset of at most one triple. (We will also assume that each point from V is used in some triple.) A Steiner triple trade {T 1 , T 2 } is an (unordered) pair of partial Steiner triple systems such that for each {x, y, z} ∈ T 1 (respectively, T 2 ), there exist

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“…These were generated using an algorithm described in [3], modified to use efficiency techniques discussed above. The entries up through n=22 were previously computed by Holton, Manvel, and McKay [25]. …”

Section: Generating Simplicial Polyhedramentioning

confidence: 99%

Polyhedra of Small Order and Their Hamiltonian Properties

Dillencourt

1996

Journal of Combinatorial Theory, Series B

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We describe the results of an enumeration of several classes of polyhedra. The enumerated classes include polyhedra with up 13 vertices, simplicial polyhedra with up to 16 vertices, 4-connected polyhedra with up to 15 vertices, non-Hamiltonian polyhedra with up to 15 vertices, bipartite polyhedra with up to 24 vertices, and bipartite trivalent polyhedra with up to 44 vertices. The results of the enumeration were used to systematically search for certain smallest non-Hamiltonian polyhedral graphs. In particular, the smallest non-Hamiltonian planar graphs satisfying certain toughness-like properties are presented here, as are the smallest non-Hamiltonian, 3-connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest non-Hamiltonian, inscribable polyhedra are also given.

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Hamiltonian cycles in cubic 3-connected bipartite planar graphs

Cited by 58 publications

References 12 publications

Supereulerian graphs: A survey

Supereulerian graphs: A survey

Planar Eulerian triangulations are equivalent to spherical Latin bitrades

Polyhedra of Small Order and Their Hamiltonian Properties

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