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4-Connected Projective-Planar Graphs Are Hamiltonian

R. Thomas, X.X. Yu
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 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.  ... 
doi:10.1006/jctb.1994.1058 fatcat:f2jsc2zkijfkhhsf4skqer4c6m

Hamiltonian cycles in 4-connected planar and projective planar triangulations with few 4-separators [article]

On-Hei Solomon Lo, Jianguo Qian
2021 arXiv   pre-print
We prove that every 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] ).  ... 
arXiv:2104.12481v1 fatcat:hktsdtdlk5e7lgbtpcguwpgsfi

Hamiltonian cycles in annular decomposable Barnette graphs [article]

Saptarshi Bej
2020 arXiv   pre-print
Barnette's conjecture is an unsolved problem in 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.  ... 
arXiv:2008.06671v1 fatcat:64zg2x3xlbcuhckqm72v55hlxy

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  ... 

Spanning trees with nonseparating paths [article]

Cristina G. Fernandes, César Hernández-Vélez, Orlando Lee, José C. de Pina
2014 arXiv   pre-print
We show that, for 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.  ... 
arXiv:1409.4239v1 fatcat:46ip7zonefezxfct4galg2o774

Uniquely and faithfully embeddable projective-planar triangulations

Seiya Negami
1984 Journal of combinatorial theory. Series B (Print)  
Classically, Whitney showed in (51 that every 3-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  ... 
doi:10.1016/0095-8956(84)90024-8 fatcat:rd7jsroejvgdbjqwit6c2nlvse

Graph-theoretical conditions for inscribability and Delaunay realizability

Michael B. Dillencourt, Warren D. Smith
1996 Discrete Mathematics  
These results have several consequences: • All 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.  ... 
doi:10.1016/0012-365x(95)00276-3 fatcat:rcsqpuezeffhloyk5tsufujesa

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.  ... 

Hamiltonicity and colorings of arrangement graphs

Stefan Felsner, Ferran Hurtado, Marc Noy, Ileana Streinu
2006 Discrete Applied Mathematics  
In this paper we show that they provide well-structured examples of families of 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.  ... 
doi:10.1016/j.dam.2006.04.006 fatcat:ogbotp5h75a65bzqfuo23vclai

Barnette's Conjecture [article]

Lean Arts, Meike Hopman, Veerle Timmermans
2013 arXiv   pre-print
Barnette in 1969 (which is an open problem in 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  ... 
arXiv:1310.5504v1 fatcat:47ongrfx2ncz5jke4pom7ipo54

The Minimality of the Georges-Kelmans Graph [article]

Gunnar Brinkmann, Jan Goedgebeur, Brendan D. McKay
2021 arXiv   pre-print
In 1969, Barnette gave a weaker version of the conjecture stating that 3-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.  ... 
arXiv:2101.00943v1 fatcat:26gn3a475rgqzmbc3grfx77m2q

Cycles in 5-connected triangulations

A. Alahmadi, R.E.L. Aldred, C. Thomassen
2019 Journal of combinatorial theory. Series B (Print)  
General rights Copyright and moral rights for the publications made accessible in the public portal 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.  ... 
doi:10.1016/j.jctb.2019.04.005 fatcat:b3kulkiu6vekfjlnjl3rt47uiu

Graphs with few Hamiltonian Cycles [article]

Jan Goedgebeur, Barbara Meersman, Carol T. Zamfirescu
2018 arXiv   pre-print
We verify up to order 48 Cantoni's conjecture that every 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.  ... 
arXiv:1812.05650v1 fatcat:stf5hqyiz5gw5gjbbqz2sy66wi

On planar regular graphs degree three without Hamiltonian cycles [article]

Emanuels Grinbergs
2009 arXiv   pre-print
Necessary condition to have 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.  ... 
arXiv:0908.2563v1 fatcat:s7evuozyrbgndc24xe3xwqmrgi

Chords of longest circuits in locally planar graphs

Ken-ichi Kawarabayashi, Jianbing Niu, Cun-Quan Zhang
2007 European journal of combinatorics (Print)  
This conjecture is verified for locally 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.  ... 
doi:10.1016/j.ejc.2005.07.017 fatcat:ja2yqxyotbcblo4icz4ia2bldi
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