scholarly journals On Cayley Digraphs That Do Not Have Hamiltonian Paths

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Dave Witte Morris
Keyword(s):  
Finite Group ◽  
Hamiltonian Path ◽  
Infinite Family ◽  
Hamiltonian Paths ◽  
Cayley Digraph

We construct an infinite family {Cay→(Gi;ai;bi)} of connected, 2-generated Cayley digraphs that do not have hamiltonian paths, such that the orders of the generators ai and bi are unbounded. We also prove that if G is any finite group with |[G,G]|≤3, then every connected Cayley digraph on G has a hamiltonian path (but the conclusion does not always hold when |[G,G]|=4 or 5).

scholarly journals Zassenhaus conjecture on torsion units holds for SL(2, 𝑝) and SL(2, 𝑝2)

2019 ◽  
Vol 22 (5) ◽  
pp. 953-974
Author(s):  
Ángel del Río ◽  
Mariano Serrano
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Infinite Family ◽  
Solvable Groups ◽  
Integral Group ◽  
Rational Group ◽  
Torsion Unit ◽  
Zassenhaus Conjecture ◽  
A Finite Group ◽  
Torsion Units

Abstract H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring {\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra {\mathbb{Q}G} to an element of G. We prove the Zassenhaus conjecture for the groups {\mathrm{SL}(2,p)} and {\mathrm{SL}(2,p^{2})} with p a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus conjecture has been proved. We also prove that if {G=\mathrm{SL}(2,p^{f})} , with f arbitrary and u is a torsion unit of {\mathbb{Z}G} with augmentation 1 and order coprime with p, then u is conjugate in {\mathbb{Q}G} to an element of G. By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of {\mathbb{Z}G} of order a multiple of p and augmentation 1 has order actually equal to p.

Every finite group action on a compact 3-manifold preserves infinitely many hyperbolic spatial graphs

2014 ◽  
Vol 23 (06) ◽  
pp. 1450034 ◽  
Author(s):  
Toru Ikeda
Keyword(s):  
Finite Group ◽  
Group Action ◽  
Group Actions ◽  
Infinite Family ◽  
Finite Group Action ◽  
Finite Group Actions ◽  
Spatial Graph ◽  
Cell Decompositions ◽  
Spatial Graphs ◽  
Ideal Triangulations

We consider symmetries of spatial graphs in compact 3-manifolds described by smooth finite group actions. This paper provides a method for constructing an infinite family of hyperbolic spatial graphs with given symmetry by connecting spatial graph versions of hyperbolic tangles in 3-cells of polyhedral cell decompositions induced from triangulations of the 3-manifolds. This method is applicable also to the case of ideal triangulations.

scholarly journals New Sufficient Conditions for Hamiltonian Paths

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
M. Sohel Rahman ◽  
M. Kaykobad ◽  
Jesun Sahariar Firoz
Keyword(s):  
Hamiltonian Path ◽  
Sufficient Conditions ◽  
Hamiltonian Paths ◽  
Hamiltonian Path Problem

A Hamiltonian path in a graph is a path involving all the vertices of the graph. In this paper, we revisit the famous Hamiltonian path problem and present new sufficient conditions for the existence of a Hamiltonian path in a graph.

scholarly journals Hamiltonian Paths in Some Classes of Grid Graphs

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Fatemeh Keshavarz-Kohjerdi ◽  
Alireza Bagheri
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hamiltonian Paths ◽  
Grid Graphs ◽  
Hamiltonian Path Problem ◽  
Np Complete ◽  
Necessary And Sufficient ◽  
Linear Time Algorithms

The Hamiltonian path problem for general grid graphs is known to be NP-complete. In this paper, we give necessary and sufficient conditions for the existence of Hamiltonian paths inL-alphabet,C-alphabet,F-alphabet, andE-alphabet grid graphs. We also present linear-time algorithms for finding Hamiltonian paths in these graphs.

scholarly journals Improving Multivariate Microaggregation through Hamiltonian Paths and Optimal Univariate Microaggregation

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 916
Author(s):  
Armando Maya-López ◽  
Fran Casino ◽  
Agusti Solanas
Keyword(s):  
Location Privacy ◽  
Hamiltonian Path ◽  
Real Life ◽  
Personal Data ◽  
Heuristic Solution ◽  
Hamiltonian Paths ◽  
Life Trajectories ◽  
Individual Privacy ◽  
Benchmark Datasets ◽  
The Traveling Salesman Problem

The collection of personal data is exponentially growing and, as a result, individual privacy is endangered accordingly. With the aim to lessen privacy risks whilst maintaining high degrees of data utility, a variety of techniques have been proposed, being microaggregation a very popular one. Microaggregation is a family of perturbation methods, in which its principle is to aggregate personal data records (i.e., microdata) in groups so as to preserve privacy through k-anonymity. The multivariate microaggregation problem is known to be NP-Hard; however, its univariate version could be optimally solved in polynomial time using the Hansen-Mukherjee (HM) algorithm. In this article, we propose a heuristic solution to the multivariate microaggregation problem inspired by the Traveling Salesman Problem (TSP) and the optimal univariate microaggregation solution. Given a multivariate dataset, first, we apply a TSP-tour construction heuristic to generate a Hamiltonian path through all dataset records. Next, we use the order provided by this Hamiltonian path (i.e., a given permutation of the records) as input to the Hansen-Mukherjee algorithm, virtually transforming it into a multivariate microaggregation solver we call Multivariate Hansen-Mukherjee (MHM). Our intuition is that good solutions to the TSP would yield Hamiltonian paths allowing the Hansen-Mukherjee algorithm to find good solutions to the multivariate microaggregation problem. We have tested our method with well-known benchmark datasets. Moreover, with the aim to show the usefulness of our approach to protecting location privacy, we have tested our solution with real-life trajectories datasets, too. We have compared the results of our algorithm with those of the best performing solutions, and we show that our proposal reduces the information loss resulting from the microaggregation. Overall, results suggest that transforming the multivariate microaggregation problem into its univariate counterpart by ordering microdata records with a proper Hamiltonian path and applying an optimal univariate solution leads to a reduction of the perturbation error whilst keeping the same privacy guarantees.

An infinite family of finite 3-groups with deficiency zero

2019 ◽  
Vol 18 (07) ◽  
pp. 1950121
Author(s):  
Reza Sabzchi ◽  
Hossein Abdolzadeh
Keyword(s):  
Finite Group ◽  
Infinite Family ◽  
Nilpotency Class

In this paper, for any integer [Formula: see text] we introduce a non-metacyclic finite group of order [Formula: see text] and nilpotency class [Formula: see text] with zero deficiency.

scholarly journals Hamiltonian paths in odd graphs

2009 ◽  
Vol 3 (2) ◽  
pp. 386-394 ◽  
Author(s):  
Letícia Bueno ◽  
Luerbio Faria ◽  
Figueiredo De ◽  
Fonseca Da
Keyword(s):  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Transitive Graph ◽  
Direct Computation ◽  
Hamiltonian Paths ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs ◽  
Connected Vertex

Lov?sz conjectured that every connected vertex-transitive graph has a Hamiltonian path. The odd graphs Ok form a well-studied family of connected, k-regular, vertex-transitive graphs. It was previously known that Ok has Hamiltonian paths for k ? 14. A direct computation of Hamiltonian paths in Ok is not feasible for large values of k, because Ok has (2k - 1, k - 1) vertices and k/2 (2k - 1, k - 1) edges. We show that Ok has Hamiltonian paths for 15 ? k ? 18. Instead of directly running any heuristics, we use existing results on the middle levels problem, therefore further relating these two fundamental problems, namely finding a Hamiltonian path in the odd graph and finding a Hamiltonian cycle in the corresponding middle levels graph. We show that further improved results for the middle levels problem can be used to find Hamiltonian paths in Ok for larger values of k.

scholarly journals FINITELY ANNIHILATED GROUPS

2014 ◽  
Vol 90 (3) ◽  
pp. 404-417 ◽  
Author(s):  
MAURICE CHIODO
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Infinite Family ◽  
Finitely Generated ◽  
Perfect Group ◽  
Finitely Generated Groups ◽  
Finitely Presented

AbstractIn 1976, Wiegold asked if every finitely generated perfect group has weight 1. We introduce a new property of groups, finitely annihilated, and show that this might be a possible approach to resolving Wiegold’s problem. For finitely generated groups, we show that in several classes (finite, solvable, free), being finitely annihilated is equivalent to having noncyclic abelianisation. However, we also construct an infinite family of (finitely presented) finitely annihilated groups with cyclic abelianisation. We apply our work to show that the weight of a nonperfect finite group, or a nonperfect finitely generated solvable group, is the same as the weight of its abelianisation. This recovers the known partial results on the Wiegold problem: a finite (or finitely generated solvable) perfect group has weight 1.

MUTUALLY INDEPENDENT HAMILTONIAN PATHS AND CYCLES IN HYPERCUBES

2006 ◽  
Vol 07 (02) ◽  
pp. 235-255 ◽  
Author(s):  
CHAO-MING SUN ◽  
CHENG-KUAN LIN ◽  
HUA-MIN HUANG ◽  
LIH-HSING HSU
Keyword(s):  
Bipartite Graph ◽  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Hamiltonian Cycles ◽  
Hamiltonian Paths ◽  
Mutually Independent ◽  
Hamiltonian Laceability

Two hamiltonian paths P1 = 〈v1, v2, …, vn(G) 〉 and P2 = 〈 u1, u2, …, un(G) 〉 of G are independent if v1 = u1, vn(G) = un(G), and vi ≠ ui for 1 < i < n(G). A set of hamiltonian paths {P1, P2, …, Pk} of G are mutually independent if any two different hamiltonian paths in the set are independent. A bipartite graph G is hamiltonian laceable if there exists a hamiltonian path joining any two nodes from different partite sets. A bipartite graph is k-mutually independent hamiltonian laceable if there exist k-mutually independent hamiltonian paths between any two nodes from different partite sets. The mutually independent hamiltonian laceability of bipartite graph G, IHPL(G), is the maximum integer k such that G is k-mutually independent hamiltonian laceable. Let Qn be the n-dimensional hypercube. We prove that IHPL(Qn) = 1 if n ∈ {1,2,3}, and IHPL(Qn) = n - 1 if n ≥ 4. A hamiltonian cycle C of G is described as 〈 u1, u2, …, un(G), u1 〉 to emphasize the order of nodes in C. Thus, u1 is the beginning node and ui is the i-th node in C. Two hamiltonian cycles of G beginning at u, C1 = 〈 v1, v2, …, vn(G), v1 〉 and C2 = 〈 u1, u2, …, un(G), u1 〉, are independent if u = v1 = u1, and vi ≠ ui for 1 < i ≤ n(G). A set of hamiltonian cycles {C1, C2, …, Ck} of G are mutually independent if any two different hamiltonian cycles are independent. The mutually independent hamiltonianicity of graph G, IHC(G), is the maximum integer k such that for any node u of G there exist k-mutually independent hamiltonian cycles of G starting at u. We prove that IHC(Qn) = n - 1 if n ∈ {1,2,3} and IHC(Qn) = n if n ≥ 4.

scholarly journals Hamiltonian Paths in the Complete Graph with Edge-Lengths 1, 2, 3

10.37236/316 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Stefano Capparelli ◽  
Alberto Del Fra
Keyword(s):  
Complete Graph ◽  
Hamiltonian Path ◽  
Positive Answer ◽  
Hamiltonian Paths ◽  
Modulo P

Marco Buratti has conjectured that, given an odd prime $p$ and a multiset $L$ containing $p-1$ integers taken from $\{1,\ldots,(p-1)/2\}$, there exists a Hamiltonian path in the complete graph with $p$ vertices whose multiset of edge-lengths is equal to $L$ modulo $p$. We give a positive answer to this conjecture in the case of multisets of the type $\{1^a,2^b,3^c\}$ by completely classifying such multisets that are linearly or cyclically realizable.

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