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On Barnette's conjecture
2010
Discrete Mathematics
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Barnette's conjecture is the statement that every cubic 3-connected bipartite planar graph is Hamiltonian. ...
the same face, there is a Hamilton cycle through one and avoiding the other. (4) If any two edges are chosen which are an even distance apart on the same face, there is a Hamilton cycle which avoids both ...
Kelmans [8] proved that Barnette's conjecture holds if and only if for any graph in P and for any two edges belonging to the same face of this graph, there is a Hamilton cycle through one and avoiding ...
doi:10.1016/j.disc.2010.01.018
fatcat:wpgq3dftkng4dlamfp4xy6wete
Thoughts on Barnette's Conjecture
[article]
2013
arXiv
pre-print
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We also explain related results on Barnette's Conjecture that were obtained by Kelmans and for which detailed self-contained proofs have not been published. ...
Our final result highlights the limitations of using a proper colouring of G as a starting point for proving Barnette's Conjecture. ...
One important result due to Kelmans [15] establishes equivalence between Barnette's Conjecture and several apparently different statements. ...
arXiv:1312.3783v1
fatcat:lga6pvliobgrjbcfo4wigecyde
Remarks on Barnette's conjecture
2019
Journal of combinatorial optimization
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Holton et al. (1985) have used computer search to confirm Barnette's conjecture for graphs up to 64 vertices. ...
In 1969, Barnette (Tutte 1969, Problem 5) conjectured that every graph in P has a Hamilton cycle. ...
doi:10.1007/s10878-019-00460-8
fatcat:qejshdsclzasrf6lef2taehvzi
Remarks on Barnette's Conjecture
[article]
2018
arXiv
pre-print
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Holton, Manvel and McKay [6] used computer search to confirm Barnette's conjecture for graphs up to 64 vertices. ...
Barnette, in 1969 ([9] , Problem 5), conjectured that every graph in P has a Hamilton cycle. ...
arXiv:1807.08933v2
fatcat:auwdxv7rtreafp3xe6tzscgfte
Thoughts on Barnette's Conjecture
2016
The Australasian Journal of Combinatorics
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This result implies the following special case of Barnette's Conjecture: if G is an Eulerian planar triangulation, whose vertices are properly ...
One important result due to Kelmans [14] establishes equivalence between Barnette's Conjecture and several apparently different statements. ...
In the following section we will see that the most important graphs for Barnette's conjecture are those without separating triangles. ...
dblp:journals/ajc/AltPSW16
fatcat:hiospcxi5zahbmcchohy53wzn4
On Barnette's Conjecture and H^+- property
[article]
2012
arXiv
pre-print
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A conjecture of Barnette states that every 3-connected cubic bipartite plane graph has a Hamilton cycle, which is equivalent to the statement that every simple even plane triangulation admits a partition ...
[B_1 ∪ B_3] are acyclic, then the following properties are satisfied: [6pt] (1) For every path abc there is possible to partition the vertex set of G into two subsets so that each induces a tree, and one ...
: Kelmans [7] proved that Barnette's conjecture holds if and only if every graph in P has the property H +− . ...
arXiv:1208.4332v1
fatcat:76qumtsa2zcnnj63k6roebbfx4
A note on Barnette's conjecture
2011
Discrete Mathematics
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In this paper we present a new approach to Barnette's conjecture by using 2-tree coloring. ...
We also define extendable, non-extendable and compatible graphs; and discuss their connection with Barnette's conjecture. Published by Elsevier B.V. ...
Therefore the study of non-extendable compatible graphs will certainly shed more light on Barnette's conjecture. EFig. 1 . 1 -mail address: xiaoyunl@hotmail.com. 0012-365X/$ -see front matter. ...
doi:10.1016/j.disc.2011.08.011
fatcat:a24iwrvzajexta7icysk6xwxyq
A note on Barnette's Conjecture
2013
Discussiones Mathematicae Graph Theory
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We prove that this conjecture is equivalent to the statement that there is a constant c > 0 such that each graph G of this class contains a path on at least c|V (G)| vertices. ...
Barnette conjectured that each planar, bipartite, cubic, and 3-connected graph is hamiltonian. ...
If Barnette's Conjecture is true then, with c = 1, each Barnette graph G contains a path on at least c|V (G)| vertices. ...
doi:10.7151/dmgt.1643
fatcat:2lul2excpvbapdy4l7336mlj3u
Construction of Barnette graphs whose large subgraphs are non-Hamiltonian
2019
Acta Universitatis Sapientiae: Mathematica
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Barnette's conjecture states that every three connected cubic bipartite planar graph (CPB3C) is Hamiltonian. ...
Theorem 8 [ 9 ] 89 Barnette's conjecture is true if and only if there is a constant c > 0 such that each Barnette graph G contains a path on at least c|V(G)| vertices. ...
Theorem 5 Barnette's conjecture holds if and only if for any arbitrary face in a Barnette graph there is a Hamiltonian cycle which passes through any two arbitrary edges on that face. ...
doi:10.2478/ausm-2019-0026
fatcat:lnnlgxiknfesfptvhx6gjsy3fa
Cyclic Subsets and Barnette's Conjecture
[article]
2014
arXiv
pre-print
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2013 pre-print arXiv:1309.2560v1 fatcat:oqp6ggncznbfrb6wbw7xawbqdm
This paper uses this theorem to construct an inductive proof of Barnette's long-standing conjecture, which asks whether every cubic, polyhedral, bipartite graph is Hamiltonian. ...
Finding a class of graphs that are certain to be Hamiltonian is one of the biggest unsolved problems in Hamiltonian graph theory today. ...
Barnette's conjecture has stood unsolved since its publication 40 years ago. The primary function of this paper is to present a proof of Barnette's conjecture. ...
arXiv:1309.2560v2
fatcat:omcxdxsr2nahtoqmzmv4yrd3pm
HAMILTONICITY OF CAMOUFLAGE GRAPHS
2021
International Journal of Engineering Applied Sciences and Technology
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The well-known mathematician Barnette introduced the open conjecture which becomes a theorem by restricting our attention to the class of graphs which is 3-regular, 3- connected, bipartite, planar graphs ...
proved in this paper stated that "Every connected vertex-transitive simple graph has a Hamilton path" shows a significant improvement over the previous efforts by L.Babai and L.Lovasz who put forth this conjecture ...
A BRIEF STUDY ABOUT THE BARNETTE'S CONJECTURE In 1884, the conjecture of Tait's stated that "Every cubic planar 3-connected graph is Hamiltonian". ...
doi:10.33564/ijeast.2021.v05i10.022
fatcat:yylc75ifrng67cpwixa5m4jfti
Barnette's Conjecture
[article]
2013
arXiv
pre-print
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This report provides an overview of theorems and statements related to a conjecture stated by D.W. ...
that Barnette's conjecture is also NP-complete. ...
If we combine these two facts we conclude that Barnette's conjecture is on the boundary between NP-completeness and P. ...
arXiv:1310.5504v1
fatcat:47ongrfx2ncz5jke4pom7ipo54
Page 1602 of Mathematical Reviews Vol. , Issue 2003C
[page]
2003
Mathematical Reviews
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(MAL-MALA-IM; Kuala Lumpur); Ong, Siew-Hui (MAL-MALA-IM; Kuala Lumpur) On Barnette’s conjecture and CBP graphs with given number of Hamilton cycles. ...
Summary: “Barnette’s conjecture (BC) states that every 3- connected cubic bipartite planar graph admits a Hamilton cycle. It is shown that BC is equivalent to a conjecture stated in this pa- per. ...
Remarks on Missing Faces and Generalized Lower Bounds on Face Numbers
2009
Electronic Journal of Combinatorics
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Sharp analogues of McMullen's generalized lower bounds, and of Barnette's lower bounds, are conjectured for these families of complexes. Partial results on these conjectures are presented. ...
This gives a hierarchy of conjectures on lower bounds on face numbers, interpolating between the generalized lower bound conjecture for simplicial spheres [12] and Gal's conjecture for flag spheres ...
For i = 1, the conjectured lower bounds (for flag homology spheres) would follow from Gal's conjecture on the γ-polynomial [8, Conjecture 2.1.7]. ...
doi:10.37236/74
fatcat:gqxux4sf6jh5xatcu5fkzgqz7y
Remarks on missing faces and generalized lower bounds on face numbers
[article]
2009
arXiv
pre-print
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Sharp analogues of McMullen's generalized lower bounds, and of Barnette's lower bounds, are conjectured for these families of complexes. Some partial results on these conjectures are presented. ...
This gives a hierarchy of conjectures on lower bounds on face numbers, interpolating between the generalized lower bound conjecture for simplicial spheres [12] and Gal's conjecture for flag spheres ...
For i = 1, the conjectured lower bounds (for flag homology spheres) would follow from Gal's conjecture on the γ-polynomial [8, Conjecture 2.1.7]. ...
arXiv:0810.5487v2
fatcat:xg27m6m4nfawpl62czog7tbis4
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