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Hamiltonian Paths and Cycles in Planar Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

We examine the problem of counting the number of

doi:10.1007/978-3-642-31770-5_8
fatcat:rbyxu64yqfbhzexirm5tkwwhfu
*Hamiltonian**paths**and**Hamiltonian**cycles**in*outerplanar*graphs**and**planar**graphs*, respectively. ... Finally, we prove a 2.2134 n upper bound on the number of*Hamiltonian**cycles**in**planar**graphs*, which improves the previously best known upper bound 2.3404 n . ...*Hamiltonian**Cycles**in**Planar**Graphs**In*this section we modify the idea of the proof of Buchin et al. [2] to obtain an improved upper bound on the number of*Hamiltonian**cycles**in**planar**graphs*. ...##
###
Sufficient Condition and Algorithm for Hamiltonian in 3-Connected 3-Regular Planar Bipartite Graph

2015
*
International Journal of Computer Applications
*

A

doi:10.5120/20596-3096
fatcat:tejwu5a2lva5toxq5s47sqxipi
*graph*G (V, E) is said to be*Hamiltonian*if it contains a spanning*cycle*. The spanning*cycle*is called a*Hamiltonian**cycle*of G*and*G is said to be a*Hamiltonian**graph*. ...*In*this paper, we study Hamiltonicity of 3-connected, 3-regular*planar*bipartite*graph*G with partite sets V=M N. We shall prove that G has a*Hamiltonian**cycle*if G is balanced with M = N. ... A*Hamiltonian**cycle*is a spanning*cycle**in*a*graph*i.e. a*cycle*through every vertex*and*a*Hamiltonian**path*is a spanning*path*. A*graph*containing a*Hamiltonian**cycle*is said to be*Hamiltonian*. ...##
###
Complexity of the hamiltonian cycle in regular graph problem

1994
*
Theoretical Computer Science
*

When the k-regular

doi:10.1016/0304-3975(94)90185-6
fatcat:yz7coqv7nvc6flbe27nv6f7zki
*graph*is*planar*, deciding whether the*graph*has a*hamiltonian**cycle*(or*path*) was proved NP-complete fork'= 3*and*polynomial for k $6. ... We prove that for k = 4*and*k =)5 the problem is NP-complete. :. " Complexity of the*hamiltonian**cycle**in*regular*graph*problem A A ... For k = 4*and*k = 5, we prove that deciding whether a 4-regular*planar**graph*or a 5-regular*planar**graph*has a*hamiltonian**cycle*(or*path*) are two NP-complete problems. ...##
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Each maximal planar graph with exactly two separating triangles is Hamiltonian

2007
*
Discrete Applied Mathematics
*

Chen [Any maximal

doi:10.1016/j.dam.2007.03.018
fatcat:npbltdz3lrao5aewhboeeu7l4i
*planar**graph*with only one separating triangle is*Hamiltonian*J. Combin. ... Optim. 7 (2003) 79-86] proved that any maximal*planar**graph*with only one separating triangle is still*Hamiltonian*. ... G out*and*G*in*are both maximal*planar**graphs*with no separating triangles. By Theorem 7, G*in*is*Hamiltonian*for any two boundary edges. ...##
###
A Two-Connected Graph with Gallai's Property

2019
*
Advances in Wireless Communications and Networks
*

that whether there exists

doi:10.11648/j.awcn.20190501.14
fatcat:cems36b3e5d3jjlugdxp4htjqi
*graphs*of*Paths**and**Cycles*, that is to say i-connected*graphs*(*planar*or non-*planar*respectively), such that each set of j points are disjoint from some longest*paths*or*cycles*... The most famous examples of Hypo-*Hamiltonian**graph*is the Petersen*graph*. ... A*graph*is said to be traceable if it has a*Hamiltonian**path**and*a*graph*is said to be*Hamiltonian*if it has a*Hamiltonian**cycle*. ...##
###
On spanning subgraphs of 4-connected planar graphs

1989
*
Discrete Applied Mathematics
*

We also show that the complexity of finding such a spanning subgraph is polynomially equivalent to the complexity of finding a

doi:10.1016/0166-218x(89)90006-1
fatcat:4kbrfwjf5ffzbk6ocy7xqbj6xe
*Hamiltonian**cycle**in*a 4-connected*planar**graph*. 0166-218X/89/$3.50 0 1989 ... We show that any 4-connected*planar**graph*G contains a spanning figure-eight subgraph based at g for any vertex g of G. ... Acknowledgment The author wishes to thank the referees for their constructive comments*and*for calling his attention to Kotzig's work. ...##
###
Spanning trees with nonseparating paths
[article]

2014
*
arXiv
*
pre-print

We consider questions related to the existence of spanning trees

arXiv:1409.4239v1
fatcat:46ip7zonefezxfct4galg2o774
*in**graphs*with the property that after the removal of any*path**in*the tree the*graph*remains connected. ... The*cycle*space of a*graph*can be generated by the fundamental*cycles*of any spanning tree,*and*Tutte showed, that for a 3-connected*graph*, it can be generated by nonseparating*cycles*. ... Figure 2 : 2 3-connected nonplanar*graphs*with a Tutte tree*in*dashed edges.Unlike the*planar**graphs*, the existence of a Tutte tree*in*a nonplanar*graph*implies neither a*Hamiltonian**cycle*nor a*Hamiltonian*...##
###
Are highly connected 1-planar graphs Hamiltonian?
[article]

2019
*
arXiv
*
pre-print

It is well-known that every

arXiv:1911.02153v1
fatcat:v2w2mtwuvbe5xhamignpmimwkm
*planar*4-connected*graph*has a*Hamiltonian**cycle*.*In*this paper, we study the question whether every 1-*planar*4-connected*graph*has a*Hamiltonian**cycle*. ... We show that this is false*in*general, even for 5-connected*graphs*, but true if the*graph*has a 1-*planar*drawing where every region is a triangle. ...*Graph*G − is 4-connected*and**planar*, so we can find a*Hamiltonian**cycle**in*G − (hence also G). ...##
###
Hamiltonian cycles and 1-factors in 5-regular graphs
[article]

2021
*
arXiv
*
pre-print

On the other hand, by the Four Colour Theorem

arXiv:2008.03173v2
fatcat:5cyupj3qcreenc7y6hbss4nowe
*and*a result of Brinkmann*and*the first author, every*planar*4-connected 5-regular*graph*satisfying a condition on its*hamiltonian**cycles*has a linear number ... . form a*hamiltonian**cycle*. ... Since G has h*hamiltonian**cycles*through e*and*e ′ , H i has h*hamiltonian*x i y i -*paths*through e ′ i . ...##
###
Simultaneous Embedding of Planar Graphs with Few Bends
[chapter]

2005
*
Lecture Notes in Computer Science
*

We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-

doi:10.1007/978-3-540-31843-9_21
fatcat:335gpmzsijalddsrbxfbfw426u
*path*pairs on the O(n) × O(n 2 ) grid with at most one bend per tree-edge*and*no bends along*path*edges. ... We present an O(n) time algorithm for simultaneous embedding of pairs of*planar**graphs*on the O(n 2 )×O(n 2 ) grid, with at most three bends per edge, where n is the number of vertices. ... Acknowledgments We would like to thank Michael Kaufmann*and*David Eppstein for the interesting discussions about the problems discussed*in*the paper,*and*Petr Moravsky for helping with the implementation ...##
###
A Planar Lattice Graph, with Empty Intersection of All Longest Path

2019
*
Engineering Mathematics
*

torus, Mobius strip,

doi:10.11648/j.engmath.20190301.12
fatcat:nqb3sqzicvazbh4di42wu6sgtq
*and*the Klein bottle but no hypo-*Hamiltonian**graphs*are embeddable*in*the*planar*square lattice. ... Walther, who introduced a*planar**graph*on 25 vertices satisfying Gallai's property,*and*various authors worked on that property, after examples of such*graphs*were found while examining such n-dimensional ... Introduction A*graph*is*Hamiltonian*if there exists a*Hamiltonian**cycle**in*, i.e. a*cycle*which passes through every vertex of*graph*. ...##
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On Hypohamiltonian and Almost Hypohamiltonian Graphs

2014
*
Journal of Graph Theory
*

A

doi:10.1002/jgt.21815
fatcat:mal6l7cfe5fcdbbo5fr6zvajjq
*graph*G is*hamiltonian*(traceable) if it contains a*hamiltonian**cycle*(*hamiltonian**path*), i.e. a*cycle*(*path*) visiting every vertex of the*graph*. ... As G − a 3 is*hamiltonian*, there is a*hamiltonian**path**in*A 2 with end-vertices b 1*and*b 2 . These*paths*together with a 1 b 1*and*a 2 b 2 yield a*hamiltonian**cycle**in*G, a contradiction. ...##
###
Feedback Vertex Set on Hamiltonian Graphs
[article]

2021
*
arXiv
*
pre-print

We study the computational complexity of Feedback Vertex Set on subclasses of

arXiv:2104.05322v1
fatcat:c5qngtdgrjepnbdnijd6gmd5ju
*Hamiltonian**graphs*.*In*particular, we consider*Hamiltonian**graphs*that are regular or are*planar**and*regular. ... Moreover, we study the less known class of p-*Hamiltonian*-ordered*graphs*, which are*graphs*that admit for any p-tuple of vertices a*Hamiltonian**cycle*visiting them*in*the order given by the tuple. ... The magenta*path*depicts the*Hamiltonian**cycle*before the D-insertions,*and*the blue*path*depicts the*Hamiltonian**cycle*after the D-insertions. ...##
###
Simultaneous Embedding of Planar Graphs with Few Bends

2005
*
Journal of Graph Algorithms and Applications
*

We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-

doi:10.7155/jgaa.00113
fatcat:jnloyknqozbxzkf4a6wlmxeadq
*path*pairs on the O(n) × O(n 2 ) grid with at most one bend per tree-edge*and*no bends along*path*edges. ... We present an O(n) time algorithm for simultaneous embedding of pairs of*planar**graphs*on the O(n 2 )×O(n 2 ) grid, with at most three bends per edge, where n is the number of vertices. ... Acknowledgments We would like to thank Michael Kaufmann*and*David Eppstein for the interesting discussions about the problems discussed*in*the paper,*and*Petr Moravsky for helping with the implementation ...##
###
Planar Hamiltonian chordal graphs are cycle extendable

2002
*
Discrete Mathematics
*

*and*|V (C )| = |V (C)| + 1. A

*graph*G is

*cycle*extendable if G contains at least one

*cycle*

*and*every non-

*Hamiltonian*

*cycle*

*in*G is extendable. ... Hendry (Discrete Math. 85 (1990) 59) asked if every

*Hamiltonian*chordal

*graph*is

*cycle*extendable. We prove that every

*planar*

*Hamiltonian*chordal

*graph*is

*cycle*extendable. ... Theorem 2 . 2 Every

*planar*

*Hamiltonian*chordal

*graph*is

*cycle*extendable. Proof. Let G be a

*planar*

*Hamiltonian*chordal

*graph*. We use induction on n = |V (G)|. ...

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