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1
Content available remote A note on the approximation ratio of graph-coloring
100%
Paschos V. T.
|
2001
|
tom Vol. 26, No. 4
267-271
EN
The purpose of this note is to prove that the minimum graph coloring can be approximated in polynomial time within approximation ratio less than the maximum between O(n/loge(l n), , and 0(A loglogn/logn) where n and A are the order and the maximum degree of the input-graph. If the maximum is realized by the first quantity, then it outer-performs the best known approximation ratio - function of n - for coloring (M. M. Hallddrsson, A still better performance guarantee for approximate graph coloring), while, if it is realized by the second quantity, it outer-performs the best known approximation ratio, function of A (D. Karger, R. Motwani, and M. Sudan, Approximate graph coloring by semidefinite programming).
2
Content available Frequency planning and ramifications of coloring
100%
Eisenblätter A. , Grötschel M. , Koster A.
Discussiones Mathematicae Graph Theory
|
2002
|
tom 22
|
nr 1
51-88
EN
This paper surveys frequency assignment problems coming up in planning wireless communication services. It particularly focuses on cellular mobile phone systems such as GSM, a technology that revolutionizes communication. Traditional vertex coloring provides a conceptual framework for the mathematical modeling of many frequency planning problems. This basic form, however, needs various extensions to cover technical and organizational side constraints. Among these ramifications are T-coloring and list coloring. To model all the subtleties, the techniques of integer programming have proven to be very useful.The ability to produce good frequency plans in practice is essential for the quality of mobile phone networks. The present algorithmic solution methods employ variants of some of the traditional coloring heuristics as well as more sophisticated machinery from mathematical programming. This paper will also address this issue.Finally, this paper discusses several practical frequency assignment problems in detail, states the associated mathematical models, and also points to public electronic libraries of frequency assignment problems from practice. The associated graphs have up to several thousand vertices and range form rather sparse to almost complete.
3
Content available Generalized colorings and avoidable orientations
100%
Szigeti J. , Tuza Z.
Discussiones Mathematicae Graph Theory
|
1997
|
tom 17
|
nr 1
137-145
EN
Gallai and Roy proved that a graph is k-colorable if and only if it has an orientation without directed paths of length k. We initiate the study of analogous characterizations for the existence of generalized graph colorings, where each color class induces a subgraph satisfying a given (hereditary) property. It is shown that a graph is partitionable into at most k independent sets and one induced matching if and only if it admits an orientation containing no subdigraph from a family of k+3 directed graphs.
4
Content available The NP-completeness of automorphic colorings
80%
Mazzuoccolo G.
Discussiones Mathematicae Graph Theory
|
2010
|
tom 30
|
nr 4
705-710
EN
Given a graph G, an automorphic edge(vertex)-coloring of G is a proper edge(vertex)-coloring such that each automorphism of the graph preserves the coloring. The automorphic chromatic index (number) is the least integer k for which G admits an automorphic edge(vertex)-coloring with k colors. We show that it is NP-complete to determine the automorphic chromatic index and the automorphic chromatic number of an arbitrary graph.
5
Content available 3-consecutive c-colorings of graphs
80%
Bujtás C. , Sampathkumar E. , Tuza Z. , Subramanya M. , Dominic C.
Discussiones Mathematicae Graph Theory
|
2010
|
tom 30
|
nr 3
393-405
EN
A 3-consecutive C-coloring of a graph G = (V,E) is a mapping φ:V → ℕ such that every path on three vertices has at most two colors. We prove general estimates on the maximum number $(χ̅)_{3CC}(G)$ of colors in a 3-consecutive C-coloring of G, and characterize the structure of connected graphs with $(χ̅)_{3CC}(G) ≥ k$ for k = 3 and k = 4.
6
Content available On-line 𝓟-coloring of graphs
80%
Borowiecki P.
Discussiones Mathematicae Graph Theory
|
2006
|
tom 26
|
nr 3
389-401
EN
For a given induced hereditary property 𝓟, a 𝓟-coloring of a graph G is an assignment of one color to each vertex such that the subgraphs induced by each of the color classes have property 𝓟. We consider the effectiveness of on-line 𝓟-coloring algorithms and give the generalizations and extensions of selected results known for on-line proper coloring algorithms. We prove a linear lower bound for the performance guarantee function of any stingy on-line 𝓟-coloring algorithm. In the class of generalized trees, we characterize graphs critical for the greedy 𝓟-coloring. A class of graphs for which a greedy algorithm always generates optimal 𝓟-colorings for the property 𝓟 = K₃-free is given.
7
Content available The cost chromatic number and hypergraph parameters
80%
Bacsó G. , Tuza Z.
Discussiones Mathematicae Graph Theory
|
2006
|
tom 26
|
nr 3
369-376
EN
In a graph, by definition, the weight of a (proper) coloring with positive integers is the sum of the colors. The chromatic sum is the minimum weight, taken over all the proper colorings. The minimum number of colors in a coloring of minimum weight is the cost chromatic number or strength of the graph. We derive general upper bounds for the strength, in terms of a new parameter of representations by edge intersections of hypergraphs.
8
Content available Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph
80%
Mojdeh D.
Discussiones Mathematicae Graph Theory
|
2006
|
tom 26
|
nr 1
59-72
EN
In a given graph G = (V,E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G,c). The d(G = Cₘ × Kₙ, χ(G)) has been studied. In this note we show that the exact value of defining number d(G = Cₘ × Kₙ, c) with c > χ(G), where n ≥ 2 and m ≥ 3, unless the defining number $d(K₃×C_{2r},4)$, which is given an upper and lower bounds for this defining number. Also some bounds of defining number are introduced.
9
Content available Analogues of cliques for oriented coloring
80%
Klostermeyer W. , MacGillivray G.
Discussiones Mathematicae Graph Theory
|
2004
|
tom 24
|
nr 3
373-387
EN
We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.
10
Content available remote Choice-Perfect Graphs
80%
Tuza Z.
Discussiones Mathematicae Graph Theory
|
2013
|
tom 33
|
nr 1
231-242
EN
Given a graph G = (V,E) and a set Lv of admissible colors for each vertex v ∈ V (termed the list at v), a list coloring of G is a (proper) vertex coloring ϕ : V → S v2V Lv such that ϕ(v) ∈ Lv for all v ∈ V and ϕ(u) 6= ϕ(v) for all uv ∈ E. If such a ϕ exists, G is said to be list colorable. The choice number of G is the smallest natural number k for which G is list colorable whenever each list contains at least k colors. In this note we initiate the study of graphs in which the choice number equals the clique number or the chromatic number in every induced subgraph. We call them choice-ω-perfect and choice-χ-perfect graphs, respectively. The main result of the paper states that the square of every cycle is choice-χ-perfect.
11
Content available remote Dynamic Coloring of Graphs
60%
Borowiecki P. , Sidorowicz E.
|
2012
|
tom Vol. 114, nr 2
105-128
EN
Dynamics is an inherent feature of many real life systems so it is natural to define and investigate the properties of models that reflect their dynamic nature. Dynamic graph colorings can be naturally applied in system modeling, e.g. for scheduling threads of parallel programs, time sharing in wireless networks, session scheduling in high-speed LAN’s, channel assignment inWDM optical networks as well as traffic scheduling. In the dynamic setting of the problem, a graph we color is not given in advance and new vertices together with adjacent edges are revealed one after another at algorithm’s input during the coloring process. Moreover, independently of the algorithm, some vertices may lose their colors and the algorithm may be asked to color them again. We formally define a dynamic graph coloring problem, the dynamic chromatic number and prove various bounds on its value. We also analyze the effectiveness of the dynamic coloring algorithm Dynamic-Fit for selected classes of graphs. In particular, we deal with trees, products of graphs and classes of graphs for which Dynamic-Fit is competitive. Motivated by applications, we state the problemof dynamic coloringwith discoloring constraints for which the performance of the dynamic algorithmTime-Fit is analyzed and give a characterization of graphs k-critical for Time-Fit. Since for any fixed k > 0 the number of such graphs is finite, it is possible to decide in polynomial time whether Time-Fit will always color a given graph with at most k colors.
12
Content available Graph colorings with local constraints - a survey
60%
Tuza Z.
Discussiones Mathematicae Graph Theory
|
1997
|
tom 17
|
nr 2
161-228
EN
We survey the literature on those variants of the chromatic number problem where not only a proper coloring has to be found (i.e., adjacent vertices must not receive the same color) but some further local restrictions are imposed on the color assignment. Mostly, the list colorings and the precoloring extensions are considered. In one of the most general formulations, a graph G = (V,E), sets L(v) of admissible colors, and natural numbers $c_v$ for the vertices v ∈ V are given, and the question is whether there can be chosen a subset C(v) ⊆ L(v) of cardinality $c_v$ for each vertex in such a way that the sets C(v),C(v') are disjoint for each pair v,v' of adjacent vertices. The particular case of constant |L(v)| with $c_v$ = 1 for all v ∈ V leads to the concept of choice number, a graph parameter showing unexpectedly different behavior compared to the chromatic number, despite these two invariants have nearly the same value for almost all graphs. To illustrate typical techniques, some of the proofs are sketched.
13
State minimization by means of incompatibility graph coloring
60%
Czerwiński R. , Kania D.
|
2011
|
tom Vol. 52, nr 3
160-162
EN
A method for state minimization of FSMs is presented which is based on coloring the incompatibility graph, introduced in letter algorithm, it is very compact and can be implemented as a quick computer program, especially as a preprocessing method in the process of exact state minimization.
PL
Metoda minimalizacji stanów układów sekwencyjnych jest przedstawiona w artykule. Zaprezentowana metoda wykorzystująca metodę kolorowania grafu niezgodności stanów jest bardzo zwięzła i może być zaimplementowana jako szybki program komputerowy. Metoda jest szczególnie przydatna w procesie wstępnego zredukowania liczby stanów w celu przeprowadzenia dokładnej minimalizacji.
14
Content available remote Modele i metody kolorowania grafów. Część I
51%
Kubale M.
|
2010
|
tom R. 86, nr 9
115-117
PL
Niniejszy artykuł jest pierwszą częścią 2-odcinkowego cyklu przeglądowego na temat modeli i metod kolorowania grafów. Przedstawiono w nim najważniejsze, z punktu widzenia zastosowań, modele kolorowania grafów. W szczególności pokazano: (1) co można kolorować w grafie (np. wierzchołki, krawędzie, końcówki, ściany, jednocześnie wierzchołki i krawędzie) oraz (2) jak można kolorować (np. dzielenie kolorów, zawijanie kolorów). Ponieważ kolorowanie we wszystkich tych odmianach i wariantach jest NP-trudne, podajemy oszacowania na liczbę chromatyczną (indeks chromatyczny) oraz podajemy potencjalne zastosowania rozważanych modeli w problemach naukowo-technicznych.
EN
This is the first of a couple of review papers on models and methods of graph coloring. We present herein the main models of graph coloring from a practical point of view. In particular, we show: (1) what elements of a graph can be colored (e.g. vertices, edges, faces, incidences) and (2) how these elements can be colored (e.g. fractional coloring, circular coloring). Since graph coloring is NP-hard in various modifications and variants, we give simple bounds on the chromatic number (chromatic index) as well as we give potential applications of the chromatic methods in science and technology.
15
Content available remote Modele i metody kolorowania grafów. Część II
51%
Kubale M.
|
2012
|
tom R. 88, nr 11a
51-55
PL
Niniejszy artykuł jest drugą częścią 2-odcinkowego cyklu przeglądowego na temat modeli i metod kolorowania grafów. Przedstawiono w nim najważniejsze, z punktu widzenia zastosowań, modele kolorowania grafów. W szczególności pokazano różne kryteria i ograniczenia modyfikujące kolorowanie klasyczne. Ponieważ kolorowanie we wszystkich tych odmianach i wariantach jest NP-trudne, podano oszacowania na liczbę chromatyczną (indeks chromatyczny) oraz podano potencjalne zastosowania rozważanych modeli w problemach naukowo-technicznych.
EN
This is the second of a couple of review papers on models and methods of graph coloring. We present herein the main models of graph coloring from a practical point of view. In particular, we show various criteria and modifications of classical coloring model. Since graph coloring is NP-hard in various modifications and variants, we give simple bounds on the chromatic number (chromatic index) as well as we give potential applications of the chromatic methods in science and technology.
16
Generowanie sąsiedztwa w algorytmach lokalnych poszukiwań uporządkowanego kolorowania grafów
51%
Dereniowski D.
|
2006
|
tom z. 143
51-56
PL
Przedstawienie rozwiązań problemów kombinatorycznych w postaci permutacji daje podstawy do konstrukcji algorytmów lokalnych poszukiwań. Uporządkowane pokolorowanie grafu można zapisać w postaci permutacji wierzchołków grafu. Podstawowe operacje, prowadzące do generowania sąsiedztwa rozwiązania, to zamiana dwóch elementów lub przesunięcie elementu permutacji. W artykule wskazujemy metodę, pozwalającą na wykonanie takich operacji w czasie O(m), przy założeniu, że dane jest drzewo eliminacji wyjściowej permutacji.
EN
Representing solutions to combinatorial problems as permutations allows us to use local search algorithms for solving them. A vertex ranking of a graph can be represented by a permutation of the vertices of the graph. Basic operations for generating a neighborhood of a current solution are swapping two elements of the permutation or changing a position of an element. We show how to perform such operations in time O(m) assuming that an elimination tree of the current permutation is given.
17
Samostabilizujący się algorytm kolorowania grafów dwudzielnych i kaktusów
51%
Kosowski A. , Kuszner Ł.
|
2006
|
tom z. 143
75-81
PL
W artykule podano algorytm rozproszonego, samostabilizującego się kolorowania grafów. Rozważamy spójny system niezależnych, asynchronicznych węzłów, z których każdy posiada tylko i wyłącznie lokalną wiedzę o systemie. Bez względu na stan początkowy system powinien osiągnąć pożądany stan globalny, wykonując w każdym z węzłów algorytm dany w postaci zbioru reguł. Zgodnie z naszą wiedzą przedstawiony algorytm jest pierwszym samostabilizującym algorytmem dokładnego kolorowania grafów dwudzielnych, działającym w wielomianowej liczbie ruchów.
EN
In the paper a distributed self-stabilizing algorithm for graph coloring is given. We consider a connected system of autonomous asynchronous nodes, each of which has only local information about the system. Regardless of the initial state, the system must achieve a desirable global state by executing a set of rules assigned to each node. Our method based on spanning trees is applied to give the first (to our knowledge) self-stabilizing algorithms working in a polynomial number of moves, which color bipartite graphs with exactly two colors.
18
Zastosowanie algorytmów rojowych do kolorowania grafów
51%
Obszarski P. , Piwakowski K.
|
2006
|
tom z. 143
107-113
PL
Przedstawiamy sposób adaptacji heurystycznej metody przeszukiwania PSO (ang. Particle Swarm Optimization) do znajdowania suboptymalnych pokolorowań wierzchołkowych grafów prostych. Prezentujemy sposób przeprowadzenia eksperymentów obliczeniowych oraz ich wyniki.
EN
Adaptation of the Particle Swarm Optimization method for obtaining suboptimal vertex colorings of graphs is proposed. We present details of performed computational experiments and their results.
19
Parallel tabu search for graph coloring problem
51%
Dąbrowski J. , Kubale M.
|
2006
|
tom z. 156
87-95
EN
Tabu search is a simple, yet powerful meta-heuristic based on local search that has been often used to solve combinatorial optimization problems like the graph coloring problem. This paper presents current taxonomy of parallel tabu search algorithms and compares three parallelization techniques applied to Tabucol, a sequential TS algorithm for graph coloring. The experirnental results are based on graphs available from the DIMACS benchmark suite.
20
Efektywność algorytmów dynamicznego kolorowania grafów - zastosowania w sieciach optycznych WDM
41%
Borowiecki P. , Sidorowicz E.
|
2006
|
tom z. 144
135-142
PL
W pracy tej formułujemy problem dynamicznego kolorowania grafów, analizujemy efektywność algorytmu zachłannego First-Fit (w skrócie FF) oraz wskazujemy na jego zastosowanie w problemie przydziału długości fali w sieciach optycznych WDM. W szczególności podajemy dolne i górne oszacowania dobroci algorytmu FF. Wskazujemy istnienie klas grafów G, dla których różnica pomiędzy wartością rozwiązania generowanego przez algorytm FF(G) a wartością optymalną OPT(G) może być dowolnie duża. Z drugiej strony dowodzimy, że dla dowolnego grafu G używanego przez nas w problemie przydziału długości fali zawsze zachodzi FF(G) < 20PT(G).
EN
Within this paper we introduce a problem of dynamie graph coloring and analyze effectiveness of greedy algorithm First-Fit (FF for short). We point out an important application of a new model to wavelength assignment problem in WDM networks. In particular, we give lower and upper bounds on the performance ratio of FF. We prove that for some classes of graphs G, the difference between the solution value FF(G) and optimum value OPT(G) may be arbitrarily large. On the other hand, for all graphs, that we used in the wavelength assignment problem FF(G) < 20PT(G) holds.
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