KNOTTED HAMILTONIAN CYCLES IN LINEAR EMBEDDING OF K7 INTO ℝ3

2012 ◽  
Vol 21 (14) ◽  
pp. 1250132 ◽  
Author(s):  
YOUNGSIK HUH
Keyword(s):  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Hamiltonian Cycles ◽  
Intrinsic Properties ◽  
Linear Embedding

In 1983 Conway and Gordon proved that any embedding of the complete graph K7 into ℝ3 contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recently. In this paper, we are interested in knotted Hamiltonian cycles in linear embedding of K7. Concretely it is shown that any linear embedding of K7 contains at most three figure-8 knots.

Hamiltonian Cycles in Products of Graphs

1975 ◽  
Vol 17 (5) ◽  
pp. 763-765 ◽  
Author(s):  
Joseph Zaks
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Complete Bipartite Graph ◽  
Hamiltonian Path ◽  
Hamiltonian Cycles ◽  
Vertex Set ◽  
Cycle Path

Let V(G) and E(G) denote the vertex set and the edge set of a graph G; let Kn denote the complete graph with n vertices and let Kn, m denote the complete bipartite graph on n and m vertices. A Hamiltonian cycle (Hamiltonian path, respectively) in a graph G is a cycle (path, respectively) in G that contains all the vertices of G.

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

scholarly journals Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles

10.37236/3610 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Louis DeBiasio ◽  
Theodore Molla
Keyword(s):  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Hamiltonian Cycles ◽  
Alternate Direction ◽  
Degree Conditions

In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$, then $D$ contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph $D$ to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even $n$, if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is sharp.

scholarly journals DISTRIBUTED ALGORITHMS FOR BUILDING HAMILTONIAN CYCLES IN k-ARY n-CUBES AND HYPERCUBES WITH FAULTY LINKS

2007 ◽  
Vol 08 (03) ◽  
pp. 253-284 ◽  
Author(s):  
IAIN A. STEWART
Keyword(s):  
Distributed Algorithms ◽  
Distributed Algorithm ◽  
Hamiltonian Cycle ◽  
Sequential Algorithm ◽  
Hamiltonian Cycles ◽  
Cube Formula

We derive a sequential algorithm Find-Ham-Cycle with the following property. On input: k and n (specifying the k-ary n-cube [Formula: see text]); F, a set of at most 2n − 2 faulty links; and v , a node of [Formula: see text], the algorithm outputs nodes v + and v − such that if Find-Ham-Cycle is executed once for every node v of [Formula: see text] then the node v + (resp. v −) denotes the successor (resp. predecessor) node of v on a fixed Hamiltonian cycle in [Formula: see text] in which no link is in F. Moreover, the algorithm Find-Ham-Cycle runs in time polynomial in n and log k. We also obtain a similar algorithm for an n-dimensional hypercube with at most n − 2 faulty links. We use our algorithms to obtain distributed algorithms to embed Hamiltonian cycles k-ary n-cubes and hypercubes with faulty links; our hypercube algorithm improves on a recently-derived algorithm due to Leu and Kuo, and our k-ary n-cube algorithm is the first distributed algorithm for embedding a Hamiltonian cycle in a k-ary n-cube with faulty links.

scholarly journals Complexity of Hamiltonian Cycle Reconfiguration

Algorithms ◽  
2018 ◽  
Vol 11 (9) ◽  
pp. 140 ◽  
Author(s):  
Asahi Takaoka
Keyword(s):  
Hamiltonian Cycle ◽  
Linear Time ◽  
Bipartite Graphs ◽  
Interval Graph ◽  
Unit Interval ◽  
Chordal Graphs ◽  
Hamiltonian Cycles ◽  
Permutation Graph ◽  
Proper Subclass ◽  
Chordal Bipartite Graphs

The Hamiltonian cycle reconfiguration problem asks, given two Hamiltonian cycles C 0 and C t of a graph G, whether there is a sequence of Hamiltonian cycles C 0 , C 1 , … , C t such that C i can be obtained from C i − 1 by a switch for each i with 1 ≤ i ≤ t , where a switch is the replacement of a pair of edges u v and w z on a Hamiltonian cycle with the edges u w and v z of G, given that u w and v z did not appear on the cycle. We show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete, settling an open question posed by Ito et al. (2011) and van den Heuvel (2013). More precisely, we show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete for chordal bipartite graphs, strongly chordal split graphs, and bipartite graphs with maximum degree 6. Bipartite permutation graphs form a proper subclass of chordal bipartite graphs, and unit interval graphs form a proper subclass of strongly chordal graphs. On the positive side, we show that, for any two Hamiltonian cycles of a bipartite permutation graph and a unit interval graph, there is a sequence of switches transforming one cycle to the other, and such a sequence can be obtained in linear time.

scholarly journals Computing Complete Graph Isomorphisms and Hamiltonian Cycles from Partial Ones

2002 ◽  
Vol 35 (1) ◽  
pp. 81-93 ◽  
Author(s):  
A. Groß e ◽  
J. Rothe ◽  
G. Wechsung
Keyword(s):  
Complete Graph ◽  
Hamiltonian Cycles

scholarly journals Feasible Bases for a Polytope Related to the Hamilton Cycle Problem

2021 ◽  
Author(s):  
Thomas Kalinowski ◽  
Sogol Mohammadian
Keyword(s):  
Decision Process ◽  
Network Flow ◽  
Hamiltonian Cycle ◽  
Hamilton Cycle ◽  
Hamiltonian Cycles ◽  
Combinatorial Interpretation ◽  
Full Characterization ◽  
Computational Support ◽  
Markov Decision ◽  
Flow Polytope

We study a certain polytope depending on a graph G and a parameter β ∈ (0,1) that arises from embedding the Hamiltonian cycle problem in a discounted Markov decision process. Literature suggests a conjecture a lower bound on the proportion of feasible bases corresponding to Hamiltonian cycles in the set of all feasible bases. We make progress toward a proof of the conjecture by proving results about the structure of feasible bases. In particular, we prove three main results: (1) the set of feasible bases is independent of the parameter β when the parameter is close to one, (2) the polytope can be interpreted as a generalized network flow polytope, and (3) we deduce a combinatorial interpretation of the feasible bases. We also provide a full characterization for a special class of feasible bases, and we apply this to provide some computational support for the conjecture.

scholarly journals Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles

2019 ◽  
Vol 39 (3) ◽  
pp. 383-393
Author(s):  
Meihua Meihua ◽  
Meiling Guan ◽  
Jirimutu Jirimutu
Keyword(s):  
Positive Integer ◽  
Hamiltonian Cycle ◽  
Hamiltonian Cycles ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Edge Partition ◽  
Definition Of

We use the Katona-Kierstead definition of a Hamiltonian cycle in a uniform hypergraph. A decomposition of complete \(k\)-uniform hypergraph \(K^{(k)}_{n}\) into Hamiltonian cycles was studied by Bailey-Stevens and Meszka-Rosa. For \(n\equiv 2,4,5\pmod 6\), we design an algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of \(K^{(3)}_{n}\) into 5-cycles has been presented for all admissible \(n\leq17\), and for all \(n=4^{m}+1\) when \(m\) is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we show if \(42~|~(n-1)(n-2)\) and if there exist \(\lambda=\frac{(n-1)(n-2)}{42}\) sequences \((k_{i_{0}},k_{i_{1}},\ldots,k_{i_{6}})\) on \(D_{all}(n)\), then \(K^{(3)}_{n}\) can be decomposed into 7-cycles. We use the method of edge-partition and cycle sequence. We find a decomposition of \(K^{(3)}_{37}\) and \(K^{(3)}_{43}\) into 7-cycles.

scholarly journals Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

10.37236/9143 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Marién Abreu ◽  
John Baptist Gauci ◽  
Domenico Labbate ◽  
Giuseppe Mazzuoccolo ◽  
Jean Paul Zerafa
Keyword(s):  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Complete Bipartite Graph ◽  
Line Graph ◽  
Sufficient Conditions ◽  
Perfect Matchings ◽  
Hamiltonian Cycles ◽  
Hamiltonian Graph ◽  
Open Problems ◽  
Hamiltonian Property

A graph admitting a perfect matching has the Perfect–Matching–Hamiltonian property (for short the PMH–property) if each of its perfect matchings can be extended to a hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH–property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, (iii) a balanced complete bipartite graph with at least 100 vertices, or (iv) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.

scholarly journals Multicoloured Hamilton Cycles

10.37236/1204 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Michael Albert ◽  
Alan Frieze ◽  
Bruce Reed
Keyword(s):  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Proof Technique ◽  
Hamilton Cycles ◽  
Local Lemma

The edges of the complete graph $K_n$ are coloured so that no colour appears more than $\lceil cn\rceil$ times, where $c < 1/32$ is a constant. We show that if $n$ is sufficiently large then there is a Hamiltonian cycle in which each edge is a different colour, thereby proving a 1986 conjecture of Hahn and Thomassen. We prove a similar result for the complete digraph with $c < 1/64$. We also show, by essentially the same technique, that if $t\geq 3$, $c < (2t^2(1+t))^{-1}$, no colour appears more than $\lceil cn\rceil$ times and $t|n$ then the vertices can be partitioned into $n/t$ $t-$sets $K_1,K_2,\ldots,K_{n/t}$ such that the colours of the $n(t-1)/2$ edges contained in the $K_i$'s are distinct. The proof technique follows the lines of Erdős and Spencer's modification of the Local Lemma.

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