Creating recreational Hamiltonian cycle problems

2004 ◽  
Vol 88 (512) ◽  
pp. 215-218 ◽  
Author(s):  
Mark A. M. Lynch
Keyword(s):  
Hamiltonian Cycle ◽  
Vertex Degree ◽  
Hamiltonian Cycles ◽  
Arbitrary Size ◽  
Method Of Solution ◽  
Average Vertex

In this paper graphs that contain unique Hamiltonian cycles are introduced. The graphs are of arbitrary size and dense in the sense that their average vertex degree is greater than half the number of vertices that make up the graph. The graphs can be used to generate challenging puzzles. The problem is particularly challenging when the graph is large and the ‘method’ of solution is unknown to the solver.

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 On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

2021 ◽  
pp. 1-13
Author(s):  
V. S. Guba
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Finite Graph ◽  
Vertex Degree ◽  
Doubling Property ◽  
Generating Set ◽  
Finite Set ◽  
Standard Set ◽  
Thompson’S Group ◽  
Average Vertex

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

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 ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

2004 ◽  
Vol 14 (05n06) ◽  
pp. 677-702 ◽  
Author(s):  
V. S. GUBA
Keyword(s):  
Growth Rate ◽  
Lower Bound ◽  
Cayley Graph ◽  
Upper Bound ◽  
Growth Function ◽  
Vertex Degree ◽  
General Growth ◽  
Thompson’S Group ◽  
Average Vertex ◽  
Thompson's Group F

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of [Formula: see text].

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 A note on the edge reconstruction conjecture

1975 ◽  
Vol 12 (1) ◽  
pp. 27-30 ◽  
Author(s):  
E.F. Schmeichel
Keyword(s):  
Degree Sequence ◽  
Existence Condition ◽  
Vertex Degree ◽  
Hamiltonian Cycles ◽  
Edge Reconstruction

Let G be a graph with vertex degree sequence d1 ≤ d2 ≤ … ≤ dp It is shown that if di + dp–i+1 ≥ p for some i, then G is uniquely reconstructable from its collection of maximal (edge deleted) subgraphs. This generalizes considerably a result of Lovász. As a corollary, it is shown that Chvátal's existence condition for hamiltonian cycles implies edge reconstructability as well.

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.

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.

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