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On Barnette's conjecture

2010
*
Discrete Mathematics
*

*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 ...

##
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Thoughts on Barnette's Conjecture
[article]

2013
*
arXiv
*
pre-print

We also explain related results

arXiv:1312.3783v1
fatcat:lga6pvliobgrjbcfo4wigecyde
*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. ...##
###
Remarks on Barnette's conjecture

2019
*
Journal of combinatorial optimization
*

Holton et al. (1985) have used computer search to confirm

doi:10.1007/s10878-019-00460-8
fatcat:qejshdsclzasrf6lef2taehvzi
*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. ...##
###
Remarks on Barnette's Conjecture
[article]

2018
*
arXiv
*
pre-print

Holton, Manvel and McKay [6] used computer search to confirm

arXiv:1807.08933v2
fatcat:auwdxv7rtreafp3xe6tzscgfte
*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. ...##
###
Thoughts on Barnette's Conjecture

2016
*
The Australasian Journal of Combinatorics
*

This result implies the following special case of

dblp:journals/ajc/AltPSW16
fatcat:hiospcxi5zahbmcchohy53wzn4
*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. ...##
###
On Barnette's Conjecture and H^+- property
[article]

2012
*
arXiv
*
pre-print

A

arXiv:1208.4332v1
fatcat:76qumtsa2zcnnj63k6roebbfx4
*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 +− . ...##
###
A note on Barnette's conjecture

2011
*
Discrete Mathematics
*

In this paper we present a new approach to

doi:10.1016/j.disc.2011.08.011
fatcat:a24iwrvzajexta7icysk6xwxyq
*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. ...##
###
A note on Barnette's Conjecture

2013
*
Discussiones Mathematicae Graph Theory
*

We prove that this

doi:10.7151/dmgt.1643
fatcat:2lul2excpvbapdy4l7336mlj3u
*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. ...##
###
Construction of Barnette graphs whose large subgraphs are non-Hamiltonian

2019
*
Acta Universitatis Sapientiae: Mathematica
*

*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. ...

##
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Cyclic Subsets and Barnette's Conjecture
[article]

2014
*
arXiv
*
pre-print

This paper uses this theorem to construct an inductive proof of

arXiv:1309.2560v2
fatcat:omcxdxsr2nahtoqmzmv4yrd3pm
*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*. ...##
###
HAMILTONICITY OF CAMOUFLAGE GRAPHS

2021
*
International Journal of Engineering Applied Sciences and Technology
*

The well-known mathematician Barnette introduced the open

doi:10.33564/ijeast.2021.v05i10.022
fatcat:yylc75ifrng67cpwixa5m4jfti
*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". ...##
###
Barnette's Conjecture
[article]

2013
*
arXiv
*
pre-print

This report provides an overview of theorems and statements related to a

arXiv:1310.5504v1
fatcat:47ongrfx2ncz5jke4pom7ipo54
*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. ...##
###
Page 1602 of Mathematical Reviews Vol. , Issue 2003C
[page]

2003
*
Mathematical Reviews
*

(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
*

Sharp analogues of McMullen's generalized lower bounds, and of

doi:10.37236/74
fatcat:gqxux4sf6jh5xatcu5fkzgqz7y
*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]. ...##
###
Remarks on missing faces and generalized lower bounds on face numbers
[article]

2009
*
arXiv
*
pre-print

Sharp analogues of McMullen's generalized lower bounds, and of

arXiv:0810.5487v2
fatcat:xg27m6m4nfawpl62czog7tbis4
*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]. ...
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