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

James Oxley, Haidong Wu
2004 Graphs and Combinatorics  
This paper specifies all such 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.  ... 
doi:10.1007/s00373-004-0552-5 fatcat:zkktnksxufhrbhhv5ig2uyycuu

A non-planar version of Tutte's wheels theorem

Talmage James Reid, Haidong Wu
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.  ... 
dblp:journals/ajc/ReidW99 fatcat:xnurjjvfzbb6zmqpqiycantb2m

Obstructions for embedding cubic graphs on the spindle surface

Dan Archdeacon, C.Paul Bonnington
2004 Journal of combinatorial theory. Series B (Print)  
Call an 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  ... 
doi:10.1016/j.jctb.2004.02.001 fatcat:v3fdx77zfbatrm7afhlezjjm5i

Coloring graphs with crossings

Bogdan Oporowski, David Zhao
2009 Discrete Mathematics  
We also consider 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.  ... 
doi:10.1016/j.disc.2008.07.040 fatcat:enz6cdly75hhplibsdu6pwmvca

Hamiltonicity of 3-connected line graphs

Weihua Yang, Liming Xiong, Hongjian Lai, Xiaofeng Guo
2012 Applied Mathematics Letters  
Zhou, Every 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.  ... 
doi:10.1016/j.aml.2012.02.032 fatcat:vnp5xljyjbaqffhgvlgy7pzi3y

Connected even factors in the square of essentially 2-edge connected graphs [article]

Jan Ekstein, Baoyindureng Wu, Liming Xiong
2017 arXiv   pre-print
In this paper we prove that 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.  ... 
arXiv:1412.8709v3 fatcat:pazcbffbz5dnbgvfpo77igltfy

A Longest Cycle Version of Tutte's Wheels Theorem

Talmage James Reid, Haidong Wu
1997 Journal of combinatorial theory. Series B (Print)  
ACKNOWLEDGMENT 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.  ... 
doi:10.1006/jctb.1997.1720 fatcat:q4mpfxllhnaznigw3ozoccocy4

A planarity criterion for cubic bipartite graphs

T. Böhme, J. Harant, A. Pruchnewski, I. Schiermeyer
1998 Discrete Mathematics  
We prove that a simple finite bipartite cubic 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.  ... 
doi:10.1016/s0012-365x(98)00090-9 fatcat:7pcajzbnffgehj7k2gbh4v2fe4

Coloring graphs with crossings [article]

Bogdan Oporowski, David Zhao
2005 arXiv   pre-print
We also consider 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.  ... 
arXiv:math/0501427v1 fatcat:a4p4yl556fhtblzsuoe4widhui

On contractible and vertically contractible elements in 3-connected matroids and graphs

Haidong Wu
1998 Discrete Mathematics  
An 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.  ... 
doi:10.1016/s0012-365x(96)00385-8 fatcat:4x6nvqzbpzfxxgnt5uayce6so4

The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices

D.A Holton, B.D McKay
1988 Journal of combinatorial theory. Series B (Print)  
We show that all 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].  ... 
doi:10.1016/0095-8956(88)90075-5 fatcat:dwjredqvgzejnld3vjfk3gxskm

On graph planarity and semi-duality

Alexander Kelmans
2001 Discrete Mathematics  
These results concern strengthenings of Kuratowski's planarity criterion for quasi-4-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]  ... 
doi:10.1016/s0012-365x(00)00078-9 fatcat:vk3uz65ribg6jnzivceo5hov5u

A zero-free interval for flow polynomials of cubic graphs

Bill Jackson
2007 Journal of combinatorial theory. Series B (Print)  
Our inductive proof technique forces us to work 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.  ... 
doi:10.1016/j.jctb.2006.04.006 fatcat:awgjeczpdzbhbjafnlbkozntdy

On longest cycles in essentially 4-connected planar graphs

Igor Fabrici, Jochen Harant, Stanislav Jendrol'
2016 Discussiones Mathematicae Graph Theory  
For a cubic 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.  ... 
doi:10.7151/dmgt.1875 fatcat:onluwzbq3zb3ze4iacrfvuvmxe

Graphs with few Hamiltonian Cycles [article]

Jan Goedgebeur, Barbara Meersman, Carol T. Zamfirescu
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.  ... 
arXiv:1812.05650v1 fatcat:stf5hqyiz5gw5gjbbqz2sy66wi
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