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.

scholarly journals Absolute Differences Along Hamiltonian Paths

10.37236/5159 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Francesco Monopoli
Keyword(s):  
Arithmetic Progression ◽  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Extreme Case ◽  
Hamiltonian Paths ◽  
Additive Energy ◽  
The Absolute ◽  
Small Additive

We prove that if the vertices of a complete graph are labeled with the elements of an arithmetic progression, then for any given vertex there is a Hamiltonian path starting at this vertex such that the absolute values of the differences of consecutive vertices along the path are pairwise distinct. In another extreme case where the label set has small additive energy, we show that the graph actually possesses a Hamiltonian cycle with the property just mentioned.  These results partially solve a conjecture by Z.-W. Sun.

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 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 Ramsey Numbers of Ordered Graphs

10.37236/7816 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Martin Balko ◽  
Josef Cibulka ◽  
Karel Král ◽  
Jan Kynčl
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Positive Answer ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
Ordered Graphs ◽  
Constant Interval ◽  
Monochromatic Copy ◽  
Special Classes

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.

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.

scholarly journals On a Problem of Marco Buratti

10.37236/109 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter Horak ◽  
Alexander Rosa
Keyword(s):  
Complete Graph ◽  
Hamiltonian Paths

We consider a problem formulated by Marco Buratti concerning Hamiltonian paths in the complete graph on $Z_p$, $p$ an odd prime.

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.

GRAPH-BASED ORDERING SCHEME FOR COLOR IMAGE FILTERING

2008 ◽  
Vol 08 (03) ◽  
pp. 473-493 ◽  
Author(s):  
O. LEZORAY ◽  
C. MEURIE ◽  
A. ELMOATAZ
Keyword(s):  
Complete Graph ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Extreme Values ◽  
Color Image ◽  
Hamiltonian Path ◽  
Image Filtering ◽  
Second Step ◽  
Median Filtering ◽  
Step Algorithm

This paper presents a graph-based ordering scheme of color vectors. A complete graph is defined over a filter window and its structure is analyzed to construct an ordering of color vectors. This graph-based ordering is constructed by finding a Hamiltonian path across the color vectors of a filter window by a two-step algorithm. The first step extracts, by decimating a minimum spanning tree, the extreme values of the color set. These extreme values are considered as the infimum and the supremum of the set of color vectors. The second step builds an ordering by constructing a Hamiltonian path among the vectors of color vectors, starting from the infimum and ending at the supremum. The properties of the proposed graph-based ordering of vectors are detailed. Several experiments are conducted to assess its filtering abilities for morphological and median filtering.

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