Wyniki wyszukiwania - Biblioteka Nauki

1

On Uniquely Hamiltonian Claw-Free and Triangle-Free Graphs

100%

Seamone B.

Discussiones Mathematicae Graph Theory

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2015

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tom 35

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nr 2

207-214

EN

A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can be modified to construct triangle-free uniquely Hamiltonian graphs with minimum degree 3.

2

Matchings Extend to Hamiltonian Cycles in 5-Cube

100%

Wang F. , Zhao W.

Discussiones Mathematicae Graph Theory

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2017

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tom 38

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nr 1

217-231

EN

Ruskey and Savage asked the following question: Does every matching in a hypercube Qn for n ≥ 2 extend to a Hamiltonian cycle of Qn? Fink confirmed that every perfect matching can be extended to a Hamiltonian cycle of Qn, thus solved Kreweras’ conjecture. Also, Fink pointed out that every matching can be extended to a Hamiltonian cycle of Qn for n ∈ {2, 3, 4}. In this paper, we prove that every matching in Q5 can be extended to a Hamiltonian cycle of Q5.

3

Addendum to “Ring elements as sums of units”

100%

Lanski C. , Maróti A.

Open Mathematics

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2013

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tom 11

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nr 5

984-984

EN

We give a comment to Theorem 1.1 published in our paper “Ring elements as sums of units” [Cent. Eur. J. Math., 2009, 7(3), 395–399].

4

Ring elements as sums of units

100%

Lanski C. , Maróti A.

Open Mathematics

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2009

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tom 7

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nr 3

395-399

EN

In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph Γ(R) is a power of 2.

5

Chvátal's Condition cannot hold for both a graph and its complement

100%

Kostochka A. , West D.

Discussiones Mathematicae Graph Theory

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2006

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tom 26

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nr 1

73-76

EN

Chvátal's Condition is a sufficient condition for a spanning cycle in an n-vertex graph. The condition is that when the vertex degrees are d₁, ...,dₙ in nondecreasing order, i < n/2 implies that $d_i > i$ or $d_{n-i} ≥ n-i$. We prove that this condition cannot hold in both a graph and its complement, and we raise the problem of finding its asymptotic probability in the random graph with edge probability 1/2.

6

A note on maximal common subgraphs of the Dirac's family of graphs

100%

Bucko J. , Mihók P. , Saclé J.-F. , Woźniak M.

Discussiones Mathematicae Graph Theory

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2005

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tom 25

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nr 3

385-390

EN

Let 𝓕 ⁿ be a given set of unlabeled simple graphs of order n. A maximal common subgraph of the graphs of the set 𝓕 ⁿ is a common subgraph F of order n of each member of 𝓕 ⁿ, that is not properly contained in any larger common subgraph of each member of 𝓕 ⁿ. By well-known Dirac's Theorem, the Dirac's family 𝓓𝓕 ⁿ of the graphs of order n and minimum degree δ ≥ [n/2] has a maximal common subgraph containing Cₙ. In this note we study the problem of determining all maximal common subgraphs of the Dirac's family $𝓓 𝓕 ^{2n}$ for n ≥ 2.

7

Retraction: Clusters analysis application on transportation network

100%

Jayarathna N. D. , Jayawardene C. J.

Journal of Sustainable Development of Transport and Logistics

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2020

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tom Vol. 5, No. 1

28--36

EN

The government of Sri Lanka established several economic centres in the provinces according to the budget proposals in the year 1998. The Dambulla economic centre was the first such centre that was established on the 01st of April 1999. Thereafter, a number of economic centres were established throughout the island. However, the Dambulla main hub remained the central warehouse of vegetables in the island. This paper deals with a vehicle scheduling problem related to transportation, and investigates a method whereby a solution can be arrived at to overcome the problem using linear programming (LP). Marketing Department Logistics (MDL) Ltd needs to distribute vegetables and fruits to different provinces. Its main hub is situated near the Dambulla vegetable and fruit market, and minor hubs are situated in different provinces in Sri Lanka. The main objective of this research is building a cost minimized model which creates a suitable method for delivering vegetables and fruits from the Dambulla major hub through its minor hubs to outlets in the provinces. Hence, to optimize the cost of outbound distribution, a mathematical model has been developed by using Integer Linear Programming, and by using reliable sources to collect data. Software assistance was obtained using the LINGO 06 optimizer, Java, MS Access and MS Excel tools to solve this mathematical model. This study is based on the Dambulla economic centre. This is an initial step to bring a correct protocol to arrange a transport model to distribute the vegetables and fruits from this centre in a cost-effective way. According to this study, all districts in Sri Lanka could be divided into four clusters. At the beginning of this research, we assumed that each district contains two warehouses and three vendors. This model is flexible enough to be re-scheduled at any request. It paves the way to create a larger model for solving any type of transportation planning problem.

8

Deflection routing with Hamiltonian cycles

75%

Szcześniak I.

Studia Informatica

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2003

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tom Vol. 24, nr 4

251-257

EN

For the mixed routing algorithm running on the networks with non buffering nodes this article presents an improvement in which an Eulerian cycle is replaced with Hamiltonian cycles. The new upper bound on a data packefs end to end number of hops is eąual to or lower than the original upper bound.

PL

W artykule proponuje się ulepszenie mieszanego algorytmu trasującego dla sieci z węzłami, które nie przechowują pakietów (ang. non-buffering nodes). Ulepszenie polega na wykorzystaniu cykli Hamiltona w zamian cyklu Eulera, co sprawia, że górna granica na liczbę skoków pakietu jest mniejsza od (albo w najbardziej niekorzystnym przypadku równa) górnej granicy przed wprowadzeniem ulepszenia.

9

Graph reduction and its application to DNA sequence assembly

75%

Błazewicz J. , Kasprzak M.

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10

Graph reduction and its application to DNA sequence assembly

75%

Błażewicz J. , Kasprzak M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2008

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tom Vol. 56, nr 1

65-70

EN

The results presented here are twofold. First, a heuristic algorithm is proposed which, through removing some unnecessary arcs from a digraph, tends to reduce it into an ad joint and thus simplifies the search for a Hamiltonian cycle. Second, a heuristic algorithm for DNA sequence assembly is proposed, which uses a graph model of the problem instance, and incorporates two independent procedures of reducing the set of arcs - one of them being the former algorithm. Finally, results of tests of the assembly algorithm on parts of chromosome arm 2R of Drosophila melanogaster are presented.

11

Algorytmizacja procesu wyznaczania długości ścieżek w sieci rozbudowanej podstawą optymalizacji zagadnień transportowych

63%

Ignaciuk S. , Wawrzosek J. , Kubera E. , Baryła-Paśnik M. , Piekarski W.

Logistyka

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2015

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tom nr 5

145--154, CD1

PL

Matematyczne metody optymalizacyjne służą jako narzędzie wykorzystywane przy podejmowaniu decyzji w działalności transportowej. Graf jest powszechnie stosowany do prezentacji problemów logistycznych. Znanym zagadnieniem w literaturze przedmiotu jest wyznaczanie najkrótszej ścieżki w grafie nie-skierowanym o nieujemnych wagach na krawędziach pomiędzy dwoma węzłami tej sieci. Odmianą tego zagadnienia jest problem znalezienia najkrótszych ścieżek pomiędzy wszystkimi parami węzłów grafu. Działanie takie w zakresie zagadnień transportowych ma na celu utworzenie pełnej macierzy odległości między wszystkimi węzłami grafu (wyznaczenie najkrótszych odległości między wszystkimi punktami na-dania i odbioru towaru). Współczesne techniki informatyczne pozwalają na przechowywanie w pamięci komputera pełnych danych zawierających informację obejmującą odległość między dwoma dowolnymi węzłami sieci i ciąg węzłów opisujących tą najkrótszą ścieżkę. W oparciu o taką istniejącą bazę można dokonywać optymalizacji procesów transportowych. Celem pracy jest wskazanie algorytmu, który na bazie wag przypisanych krawędziom grafu nieskierowanego (interpretowanymi jako odległości między dwoma węzłami) wyznacza macierz najkrótszych odległości między wszystkimi węzłami grafu oraz dodatkowo wyznacza ścieżki najkrótszych połączeń. Proces ten prowadzi m.in. do zastąpienia macierzy rzadkiej przez macierzą pełną. Równocześnie proces modelowania stanowi przejście pomiędzy zagadnieniami z zakresu teorii grafów a teorii przestrzeni metrycznych. Tak utworzona macierz pełna może stanowić podstawę do tworzenia bazy pełnych cykli Hamiltona lub innych zagadnień systemów transportowych. Wstępna analiza zaproponowanego tu algorytmu etapowego wskazuje, że jest on porównywalny ze znanymi algorytmami, a w niektórych przypadkach działa nawet szybciej. Kolejnym krokiem w następnej pracy będą czynności odwrotne do opisanych powyżej. Połączenie obu procedur pozwala na kontrolę procesu tworzenia minimal-nego grafu rozpinającego wszystkie najkrótsze ścieżki, ewentualnie tworzenie drzewa rozpinającego ścieżki, w których najkrótsze połączenia różnią się od najkrótszych dróg z zadaną z góry dokładnością. Wskazano możliwość wykorzystania wyników w optymalizacji transportu w przemyśle mleczarskim.

EN

A well-known issue in the professional literature is how to determine the shortest path in an undirected graph with non-negative weighted edges between two network nodes. A variation of this issue is to find the shortest paths between all the node pairs of a graph. Modern techniques allow one to store in the computer memory complete data containing information about the distance between any two network nodes and a sequence of nodes describing the shortest path. On the basis of such existing databases optimisation of transportation processes can be undertaken. This study aims at showing an algorithm that basing on the weights of an undirected graph edges (interpreted as the distance in some node pairs) determines the shortest distance matrix between all the nodes of the graph and additionally determines paths of shortest connections. This process leads to the replacement of a sparse matrix by a full matrix. The simultaneous modelling process forms transition between issues of the graph theory and the theory of metric spaces. The full matrix formed this way may lend itself to create the base of complete Hamilton cycles or other issues in transportation systems. An initial analysis of the stage-form algorithm proposed here indicates that it is comparable with known algorithms, and in some cases it works even faster. The next step in the subsequent study will consist in operations being reversed in comparison to the ones described above. Combining both procedures will allow for controlling the process of creating a minimal graph spanning all the shortest paths, and possibly creating a graph spanning paths, whose shortest connections differ from the shortest paths with pre-determined accuracy. The possibility of using the results for optimisation of transport in the dairy industry was shown.

12

Algorytm hybrydowy wielokrotnego startu dla rozwiązywania problemu sekwencyjnego uporządkowania

51%

Widuch J. , Klyta A.

Studia Informatica

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2014

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tom Vol. 35, nr 1

29--55

PL

Problem sekwencyjnego uporządkowania (SOP) jest podobny do asymetrycznego problemu komiwojażera. Celem jest wyznaczenie w skierowanym grafie ważonym ścieżki Hamiltona o minimalnej wadze, przy dodatkowym spełnieniu relacji pierwszeństwa wierzchołków. W niniejszej pracy zaprezentowano algorytm hybrydowy wielokrotnego startu rozwiązywania problemu SOP. Algorytm ten jest połączeniem algorytmów symulowanego wyżarzania i lokalnej optymalizacji. Dodatkowo przedstawiono wyniki przeprowadzonych badań eksperymentalnych.

EN

The sequential ordering problem (SOP) is similar to the asymmetric traveling salesman problem. The goal is to find a minimum weight Hamiltonian path on a directed weighted graph satisfying precedence relationships among the vertices. In the paper, a multistart hybrid algorithm to solving SOP is presented. The algorithm based on simulated annealing algorithm and local optimization method. Apart from that results of experimental tests are presented.