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4-Connected Projective-Planar Graphs Are Hamiltonian

1994
*
Journal of combinatorial theory. Series B (Print)
*

We prove the result stated in the title (conjectured by Grünbaum), and a conjecture of Plummer that every

doi:10.1006/jctb.1994.1058
fatcat:f2jsc2zkijfkhhsf4skqer4c6m
*graph*which can be obtained from a*4*-*connected**planar**graph*by deleting two vertices is*Hamiltonian*... The proofs*are*constructive and give rise to polynomial-time algorithms. 2 ... Acknowledgements We*are*indebted to one of the referees for carefully reading the manuscript and for providing many helpful suggestions. We would also like to acknowledge that Daniel P. ...##
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Hamiltonian cycles in 4-connected planar and projective planar triangulations with few 4-separators
[article]

2021
*
arXiv
*
pre-print

We prove that every

arXiv:2104.12481v1
fatcat:hktsdtdlk5e7lgbtpcguwpgsfi
*4*-*connected**planar*or*projective**planar*triangulation with O(n)*4*-separators has exponentially many*hamiltonian*cycles. ... Whitney proved in 1931 that every*4*-*connected**planar*triangulation is*hamiltonian*. ... Note that there*are*non-*hamiltonian**4*-*connected**graphs*that*are*embedded in the double torus or in the surface obtained from the sphere by attaching three crosscaps (see [10] ). ...##
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Hamiltonian cycles in annular decomposable Barnette graphs
[article]

2020
*
arXiv
*
pre-print

Barnette's conjecture is an unsolved problem in

arXiv:2008.06671v1
fatcat:64zg2x3xlbcuhckqm72v55hlxy
*graph*theory. The problem states that every 3-regular (cubic), 3-*connected*,*planar*, bipartite (Barnette)*graph*is*Hamiltonian*. ... Noting that Spider web*graphs**are*a subclass of Annular Decomposable Barnette (ADB*graphs*)*graphs*and*are**Hamiltonian*, we study ADB*graphs*and their annular-*connected*subclass (ADB-AC*graphs*). ...*Are*all ADB-AC*graphs**Hamiltonian*? 2.*Are*all ADB*graphs**Hamiltonian*? Lemma 2 .*4*. 24 There exists ADB*graphs*that*are*not annular-*connected*. ...##
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Page 4629 of Mathematical Reviews Vol. , Issue 2000g
[page]

2000
*
Mathematical Reviews
*

set of edges A in a given

*4*-*connected**planar*or*projective**planar**graph*G, there is a*Hamiltonian*cycle of G containing all edges of A. ... In this paper the authors consider this problem assuming that the*4*-*connected**graph*G is embedded into the plane or the*projective*plane F with face-width at least five (i.e., every simple closed curve ...##
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Spanning trees with nonseparating paths
[article]

2014
*
arXiv
*
pre-print

We show that, for

arXiv:1409.4239v1
fatcat:46ip7zonefezxfct4galg2o774
*planar**graphs*, the existence of trees with this property is closely related to the Hamiltonicity of the*graph*. ... We consider questions related to the existence of spanning trees in*graphs*with the property that after the removal of any path in the tree the*graph*remains*connected*. ... Since Tutte [8] proved that every*4*-*connected**planar**graph*is*Hamiltonian*, the corollary follows. ...##
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Uniquely and faithfully embeddable projective-planar triangulations

1984
*
Journal of combinatorial theory. Series B (Print)
*

Classically, Whitney showed in (51 that every 3-

doi:10.1016/0095-8956(84)90024-8
fatcat:rd7jsroejvgdbjqwit6c2nlvse
*connected**planar**graph*is uniquely and faithfully embeddable in a sphere, which is well known as the uniqueness of a dual of a 3-*connected**planar**graph*. ... Recently, he has shown in [3] that if a 5-*connected**projective*-*planar**graph*contains a subdivision of the complete*graph*K, with 6 vertices as its proper subgraph then it is uniquely and faithfully embeddable ...##
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Graph-theoretical conditions for inscribability and Delaunay realizability

1996
*
Discrete Mathematics
*

These results have several consequences: • All

doi:10.1016/0012-365x(95)00276-3
fatcat:rcsqpuezeffhloyk5tsufujesa
*4*-*connected*polyhedra*are*of inscribable type. • All simplicial polyhedra in which all vertex degrees*are*between*4*and 6, inclusive,*are*of inscribable type ... We present new*graph*-theoretical conditions for polyhedra of inscribable type and Delaunay triangulations. ... Since any*4*-*connected**planar**graph*is 1-*Hamiltonian*, it follows that G is k-*Hamiltonian*. ...##
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Page 9573 of Mathematical Reviews Vol. , Issue 2004m
[page]

2004
*
Mathematical Reviews
*

The work under review strengthens the

*connectivity*hypothesis to obtain the following theorem. Theorem. Every*projective*-*planar*3-*connected**graph*is*planar*. ... Suppose a*connected**graph*G admits a covering by a*projective*-*planar*G. Then G is*projective*-*planar*. Negami demonstrated that the conjecture is true if the covering*projection*is a double covering. ...##
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Hamiltonicity and colorings of arrangement graphs

2006
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Discrete Applied Mathematics
*

In this paper we show that they provide well-structured examples of families of

doi:10.1016/j.dam.2006.04.006
fatcat:ogbotp5h75a65bzqfuo23vclai
*planar*and*projective*-*planar**graphs*with very interesting properties. ... Most prominently, spherical arrangements admit decompositions into two Hamilton cycles; this is a new addition to the relatively few families of*4*-regular*graphs*that*are*known to have*Hamiltonian*decompositions ... Thomas and Yu's theorem on*4*-*connected**projective*-*planar**graphs*[24] implies a similar result for*projective*arrangements. Theorem 9. Every*projective*arrangement*graph*is*Hamiltonian*. ...##
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Barnette's Conjecture
[article]

2013
*
arXiv
*
pre-print

Barnette in 1969 (which is an open problem in

arXiv:1310.5504v1
fatcat:47ongrfx2ncz5jke4pom7ipo54
*graph*theory): Every cubic, bipartite, polyhedral*graph*contains a Hamilton cycle. ...*Planar**4*-*connected**graphs*In this chapter we will give a sketch of the proof that every*4*-*connected*,*planar**graph*G is*Hamiltonian*. ... In these two chapters we thus prove that bipartiteness and*planarity**are*necessary conditions in Barnette's conjecture and that*4*-*connectedness*is sufficient for a Barnette*graph*to contain a*Hamiltonian*...##
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The Minimality of the Georges-Kelmans Graph
[article]

2021
*
arXiv
*
pre-print

In 1969, Barnette gave a weaker version of the conjecture stating that 3-

arXiv:2101.00943v1
fatcat:26gn3a475rgqzmbc3grfx77m2q
*connected**planar*bipartite cubic*graphs**are**hamiltonian*. ... We also report that a search of small non-*hamiltonian*3-*connected*bipartite cubic*graphs*did not find any with genus less than*4*. ... Theorem*4*. 1 . 41 Let G be a 3-*connected**planar*bipartite cubic*graph*with n vertices. ...##
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Cycles in 5-connected triangulations

2019
*
Journal of combinatorial theory. Series B (Print)
*

General rights Copyright and moral rights for the publications made accessible in the public portal

doi:10.1016/j.jctb.2019.04.005
fatcat:b3kulkiu6vekfjlnjl3rt47uiu
*are*retained by the authors and/or other copyright owners and it is a condition of accessing publications ... Such*graphs**are*well known to be*Hamiltonian*. Böhme, Harant and Tkáč [6] showed that in a 5-*connected**planar*triangulation on n vertices, there*are*at least 2 O(n (1/*4*) )*Hamiltonian*cycles. ... Then the*graph*G = G − F is*4*-*connected*. 2 3.*Hamiltonian*cycles in*planar*triangulations Theorem 2. ...##
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Graphs with few Hamiltonian Cycles
[article]

2018
*
arXiv
*
pre-print

We verify up to order 48 Cantoni's conjecture that every

arXiv:1812.05650v1
fatcat:stf5hqyiz5gw5gjbbqz2sy66wi
*planar*cubic 3H*graph*contains a triangle, and show that for every k that is 0 or at least*4*there exists a*planar*cyclically*4*-edge-*connected*cubic ...*graph*with exactly k*hamiltonian*cycles. ... We would like to thank Gunnar Brinkmann for providing us with an independent program for counting*hamiltonian*cycles. ...##
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On planar regular graphs degree three without Hamiltonian cycles
[article]

2009
*
arXiv
*
pre-print

Necessary condition to have

arXiv:0908.2563v1
fatcat:s7evuozyrbgndc24xe3xwqmrgi
*Hamiltonian*cycle in*planar**graph*is given. Examples of regular*planar**graphs*degree three without*Hamiltonian*cycle*are*built. ... Let us consider*planar*embedding of the*graph*G (i.e.*planar*topological*graph*according Berge, depicting G ) with definite*Hamiltonian*cycle H in it. ... Acquired*planar**graphs*, by virtue of the theorem, should be without*Hamiltonian*cycles. ...##
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Chords of longest circuits in locally planar graphs

2007
*
European journal of combinatorics (Print)
*

This conjecture is verified for locally

doi:10.1016/j.ejc.2005.07.017
fatcat:ja2yqxyotbcblo4icz4ia2bldi
*4*-*connected**planar**graphs*, that is, let N be the set of natural numbers; then there is a function h : N → N such that, for every*4*-*connected**graph*G embedded in ... Godsil, Cycle in*graphs*, Ann. Discrete Math. 27 (1985)], p. 466) that every longest circuit of a 3-*connected**graph*must have a chord. ... It is known that every*4*-*connected**graph*in the*projective*plane is*hamiltonian*[10] , so Theorem 1.2 gives a weaker result for*projective**planar**graphs*. ...
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