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### Hamiltonian Paths and Cycles in Planar Graphs [chapter]

Sudip Biswas, Stephane Durocher, Debajyoti Mondal, Rahnuma Islam Nishat
2012 Lecture Notes in Computer Science
We examine the problem of counting the number of 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.  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

Md. Khaliluzzaman, Md. Monirul Islam, Md. Monjur Hasan
2015 International Journal of Computer Applications
A 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

C. Picouleau
1994 Theoretical Computer Science
When the k-regular 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.  ...

### Each maximal planar graph with exactly two separating triangles is Hamiltonian

Guido Helden
2007 Discrete Applied Mathematics
Chen [Any maximal 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

Abdul Naeem Kalhoro, Ali Dino Jumani
2019 Advances in Wireless Communications and Networks
that whether there exists 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

Moshe Rosenfeld
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 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]

Cristina G. Fernandes, César Hernández-Vélez, Orlando Lee, José C. de Pina
2014 arXiv   pre-print
We consider questions related to the existence of spanning trees 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]

Therese Biedl
2019 arXiv   pre-print
It is well-known that every 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]

Nico Van Cleemput, Carol T. Zamfirescu
2021 arXiv   pre-print
On the other hand, by the Four Colour Theorem 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]

Cesim Erten, Stephen G. Kobourov
2005 Lecture Notes in Computer Science
We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-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

Abdul Naeem Kalhoro, Ali Dino Jumani
2019 Engineering Mathematics
torus, Mobius strip, 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 .  ...

### On Hypohamiltonian and Almost Hypohamiltonian Graphs

Carol T. Zamfirescu
2014 Journal of Graph Theory
A 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]

Dario Cavallaro, Till Fluschnik
2021 arXiv   pre-print
We study the computational complexity of Feedback Vertex Set on subclasses of 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

Cesim Erten, Stephen G. Kobourov
2005 Journal of Graph Algorithms and Applications
We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-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

Tao Jiang
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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