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The 3-Connected Graphs with Exactly Three Non-Essential Edges

2004
*
Graphs and Combinatorics
*

This paper specifies all such

doi:10.1007/s00373-004-0552-5
fatcat:zkktnksxufhrbhhv5ig2uyycuu
*graphs**with**exactly**three**non*-*essential**edges*. ... Tutte's Wheels Theorem proves that*the*only simple*3*-*connected**graphs**with*no*non*-*essential**edges*are*the*wheels. ... Acknowledgements*The*first author was partially supported by grants from*the*National Security Agency. Thge second author was partially supported by*the*Office of Naval Research under Grant No. ...##
###
A non-planar version of Tutte's wheels theorem

1999
*
The Australasian Journal of Combinatorics
*

*The*

*graph*given in Figure 1 is a minimally

*3*-

*connected*

*non*-planar

*graph*

*with*only

*the*6

*edges*not appearing in triangles being

*non*-

*essential*. ...

*The*wheel

*graphs*are

*the*fundamental building blocks of

*graphs*[1]. Thtte's Wheels Theorem [7] characterizes

*the*wheels as being

*the*minimally

*3*-

*connected*

*graphs*

*with*no

*non*-

*essential*

*edges*. ... Hence every minimally

*3*-

*connected*

*graph*

*with*

*exactly*5

*non*-

*essential*

*edges*is a member of S. ...

##
###
Obstructions for embedding cubic graphs on the spindle surface

2004
*
Journal of combinatorial theory. Series B (Print)
*

Call an

doi:10.1016/j.jctb.2004.02.001
fatcat:v3fdx77zfbatrm7afhlezjjm5i
*edge*of e of a*non*-planar*graph*G*essential*if it is in every Kuratowski subgraph of G; that is, if and only if G À e is planar. A*graph*is ... Kuratowski's Theorem also characterizes those*graphs*that are one*edge*-deletion away from planarity: if a*graph*is*non*-planar, but*the*deletion of any*edge*makes*the**graph*planar, then*the**graph*is a subdivision ... These give*the**graphs*T 7 ; T 8 and T 9 of There are*exactly*four*three*-*edge*-*connected*obstructions G to nearplanarity*with*a*non*-trivial*three*-*edge*-cut where both sides are planar, but one side has*the*...##
###
Coloring graphs with crossings

2009
*
Discrete Mathematics
*

We also consider

doi:10.1016/j.disc.2008.07.040
fatcat:enz6cdly75hhplibsdu6pwmvca
*the*question of whether*the*result can be extended to*graphs**with*more crossings. ... We generalize*the*Five Color Theorem by showing that it extends to*graphs**with*two crossings. ...*The*crossing number of*the*complete*graph*K 6 is*three*. Proof. It is easy to draw K 6*with**exactly**three*crossings, while Proposition 1.4 implies that ν(K 6 ) ≥*3*. ...##
###
Hamiltonicity of 3-connected line graphs

2012
*
Applied Mathematics Letters
*

Zhou, Every

doi:10.1016/j.aml.2012.02.032
fatcat:vnp5xljyjbaqffhgvlgy7pzi3y
*3*-*connected*,*essentially*11-*connected*line*graph*is Hamiltonian, J. Combin. Theory Ser. B 96 (2006) 571-576] that every*3*-*connected*,*essentially*4-*connected*line*graph*is Hamiltonian. ... In this note, we first show that*the*conjecture posed by Lai et al. is not true and there is an infinite family of counterexamples; we show that*3*-*connected*,*essentially*4-*connected*line*graph*of a*graph*... Let G be a reduced*3*-*edge*-*connected**non*-trivial*graph*. Then d*3*≥ 10. Lemma*3*. 2 . 2 Let G be a*3*-*edge*-*connected**graph*. If L(G) is*essentially*4-*connected*, then L(G) is 4-*connected*. ...##
###
Connected even factors in the square of essentially 2-edge connected graphs
[article]

2017
*
arXiv
*
pre-print

In this paper we prove that

arXiv:1412.8709v3
fatcat:pazcbffbz5dnbgvfpo77igltfy
*the*square of an*essentially*2-*edge**connected**graph**with*an additional property has a*connected*even factor*with*maximum degree at most 4. ... Moreover we show that, in general,*the*square of*essentially*2-*edge**connected**graph*does not contain a*connected*even factor*with*bounded maximum degree. ...*The*first author was supported by project GA14-19503S of*the*Grant Agency of*the*Czech Republic. ...##
###
A Longest Cycle Version of Tutte's Wheels Theorem

1997
*
Journal of combinatorial theory. Series B (Print)
*

ACKNOWLEDGMENT

doi:10.1006/jctb.1997.1720
fatcat:q4mpfxllhnaznigw3ozoccocy4
*The*authors thank James G. Oxley for helpful discussions on*the*paper. ...*The**graph*G has*exactly**three**non*-*essential**edges*if and only if G is a split-wheel or a crossed splitwheel. ... Let G be a minimally*3*-*connected**graph*which is not a wheel. Then G has at least*three**non*-*essential**edges*. ...##
###
A planarity criterion for cubic bipartite graphs

1998
*
Discrete Mathematics
*

We prove that a simple finite bipartite cubic

doi:10.1016/s0012-365x(98)00090-9
fatcat:7pcajzbnffgehj7k2gbh4v2fe4
*non*-planar*graph*contains a clean subdivision of K3.3. ...*The*proof is constructive and gives rise to a polynomial-time algorithm. ... Because of*the**3*-*connectedness*of Go(M) there are*three*disjoint paths in Go(M)*connecting*A and B. ...##
###
Coloring graphs with crossings
[article]

2005
*
arXiv
*
pre-print

We also consider

arXiv:math/0501427v1
fatcat:a4p4yl556fhtblzsuoe4widhui
*the*question of whether*the*result can be extended to*graphs**with*more crossings. ... We generalize*the*Five Color Theorem by showing that it extends to*graphs**with*two crossings. ...*The*crossing number of*the*complete*graph*K 6 is*three*. Proof. It is easy to draw K 6*with**exactly**three*crossings, while Proposition 1.4 implies that ν(K 6 ) ≥*3*. ...##
###
On contractible and vertically contractible elements in 3-connected matroids and graphs

1998
*
Discrete Mathematics
*

An

doi:10.1016/s0012-365x(96)00385-8
fatcat:4x6nvqzbpzfxxgnt5uayce6so4
*edge*e in a*3*-*connected**graph*G is contractible if*the*contraction G/e is still*3*-*connected*. ... Two matroid notions of*3*-*connectedness*will be used: vertical*3*-*connectedness*,*the*analogue of*3*-*connectedness*for*graphs*, and (Tutte)*3*-*connectedness*. ...*The*author also thanks*the*referees for their very helpful comments and suggestions for*the*revision of*the*paper. ...##
###
The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices

1988
*
Journal of combinatorial theory. Series B (Print)
*

We show that all

doi:10.1016/0095-8956(88)90075-5
fatcat:dwjredqvgzejnld3vjfk3gxskm
*3*-*connected*cubic planar*graphs*on 36 or fewer vertices are hamiltonian, thus extending results of Lederberg, ... Note Added in ProoJ A recent paper of Barnette [l] demonstrates that there is no*non*-Hamiltonian C3CP on 34 vertices.*The*methods used are not dissimilar to our own. ... ACKNOWLEDGMENT We thank Joan McKay for*the*considerable effort required to compose*the*more than 50 figures required for this paper and*the*technical report version [ll]. ...##
###
On graph planarity and semi-duality

2001
*
Discrete Mathematics
*

These results concern strengthenings of Kuratowski's planarity criterion for quasi-4-

doi:10.1016/s0012-365x(00)00078-9
fatcat:vk3uz65ribg6jnzivceo5hov5u
*connected**graphs*, for bipartite quasi-4-*connected**graphs*, and for cubic bipartite*graphs*as well as generalizations ... of matroid duality of*graphs*and strengthenings of Whitney's planarity criterion. ...*The**graph*F i in Figure i is a perfect and minimal semi-dual of G i , i = 1; 2;*3*; 4. 7. Uncited references [*3*, 6, 8, 19, 21, 47, 48] ...##
###
A zero-free interval for flow polynomials of cubic graphs

2007
*
Journal of combinatorial theory. Series B (Print)
*

Our inductive proof technique forces us to work

doi:10.1016/j.jctb.2006.04.006
fatcat:awgjeczpdzbhbjafnlbkozntdy
*with*near-cubic*graphs*, that is to say*graphs**with*minimum degree at least two and at most one vertex of degree greater then*three*. ...*The*main purpose of this paper is to remove*the*planarity hypothesis from Woodall's theorem by showing that*the*dual statement holds for both planar and*non*-planar*graphs*: if G is a cubic bridgeless*graph*... In particular, for sending me a preprint of [14] which inspired*the*inductive hypothesis of Theorem 19(b). I also thank*the*referee whose comments greatly improved*the*presentation of this paper. ...##
###
On longest cycles in essentially 4-connected planar graphs

2016
*
Discussiones Mathematicae Graph Theory
*

For a cubic

doi:10.7151/dmgt.1875
fatcat:onluwzbq3zb3ze4iacrfvuvmxe
*essentially*4-*connected*planar*graph*G, Grünbaum*with*Malkevitch, and Zhang showed that G has a cycle on at least*3*4 n vertices. ... A planar*3*-*connected**graph*G is*essentially*4-*connected*if, for any 3separator S of G, one component of*the**graph*obtained from G by removing S is a single vertex. ... Lemma*3*.*3*Let G be an*essentially*4-*connected*planar*graph*, and let a and b be*non*-adjacent*edges*of G. ...##
###
Graphs with few Hamiltonian Cycles
[article]

2018
*
arXiv
*
pre-print

*graph*

*with*

*exactly*k hamiltonian cycles. ... Our main findings, combining applications of this algorithm and existing algorithms

*with*new theoretical results, revolve around

*graphs*containing

*exactly*one hamiltonian cycle (1H) or

*exactly*

*three*hamiltonian ... We would like to thank Gunnar Brinkmann for providing us

*with*an independent program for counting hamiltonian cycles. ...

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