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Search results for: BIPARTITE GRAPH COLORING
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On incidence coloring of coloring of complete multipartite and semicubic bipartite graphs
PublicationIn the paper, we show that the incidence chromatic number of a complete k-partite graph is at most ∆+2 (i.e., proving the incidence coloring conjecture for these graphs) and it is equal to ∆+1 if and only if the smallest part has only one vertex.
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The Backbone Coloring Problem for Bipartite Backbones
PublicationLet G be a simple graph, H be its spanning subgraph and λ≥2 be an integer. By a λ -backbone coloring of G with backbone H we mean any function c that assigns positive integers to vertices of G in such a way that |c(u)−c(v)|≥1 for each edge uv∈E(G) and |c(u)−c(v)|≥λ for each edge uv∈E(H) . The λ -backbone chromatic number BBCλ(G,H) is the smallest integer k such that there exists a λ -backbone coloring c of G with backbone H satisfying...
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Interval incidence coloring of bipartite graphs
PublicationIn this paper we study the problem of interval incidence coloring of bipartite graphs. We show the upper bound for interval incidence coloring number (χii) for bipartite graphs χii≤2Δ, and we prove that χii=2Δ holds for regular bipartite graphs. We solve this problem for subcubic bipartite graphs, i.e. we fully characterize the subcubic graphs that admit 4, 5 or 6 coloring, and we construct a linear time exact algorithm for subcubic...
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Chromatic cost coloring of weighted bipartite graphs
PublicationGiven a graph G and a sequence of color costs C, the Cost Coloring optimization problem consists in finding a coloring of G with the smallest total cost with respect to C. We present an analysis of this problem with respect to weighted bipartite graphs. We specify for which finite sequences of color costs the problem is NP-hard and we present an exact polynomial algorithm for the other finite sequences. These results are then extended...
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Koala graph coloring library: an open graph coloring library for real-world applications
PublicationPomimo intensywnej pracy naukowej na polu kolorowania grafów, nie jest znana kompletna i dedykowana biblioteka programistyczna. Celem artykułu jest zaproponowanie architektury takiej biblioteki. Celem jest spełnienie oczekiwań wypływających z rzeczywistych zastosowań, w szczególności spełnienie potrzeb wydajnościowych. Zaimplementowano szereg algorytmów cheurystycznego kolorowania grafów. Przyjętym językiem programowania jest C++....
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Interval incidence graph coloring
PublicationIn this paper we introduce a concept of interval incidence coloring of graphs and survey its general properties including lower and upper bounds on the number of colors. Our main focus is to determine the exact value of the interval incidence coloring number χii for selected classes of graphs, i.e. paths, cycles, stars, wheels, fans, necklaces, complete graphs and complete k-partite graphs. We also study the complexity of the...
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Mixed graph edge coloring
PublicationW pracy rozważany jest problem kolorowania krawędzi grafu mieszanego, tj. grafu zawierającego zawiero skierowane, jak i nieskierowane krawędzie. Motywację do badań stanowią zagadnienia komunikacyjne z zakresu szeregowania zadań.
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Sum Coloring of Bipartite Graphs with Bounded Degree
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Sum coloring of bipartite graphs with bounded degree.
PublicationArtykuł poświęcony jest złożoności obliczeniowej zagadnienia sumacyjnego kolorowania grafów dwudzielnych o ograniczonym stopniu. Zawiera dowód tego, że sumacyjne kolorowanie grafów dwudzielnych stopnia mniejszego równego 5 jest NP-zupełne oraz opis wielomianowego algorytmu, który optymalnie sumacyjnie koloruje grafy dwudzielne podkubiczne.
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Parallel immune system for graph coloring
PublicationThis paper presents a parallel artificial immune system designed forgraph coloring. The algorithm is based on the clonal selection principle. Each processor operates on its own pool of antibodies and amigration mechanism is used to allow processors to exchange information. Experimental results show that migration improves the performance of the algorithm. The experiments were performed using a high performance cluster on a set...
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On greedy graph coloring in the distributed model
PublicationArtykuł traktuje o zachłannym kolorowaniu grafów w modelu rozproszonym. Zaprezentowano nowy probabilistyczny algorytm dający w wyniku pokolorowanie LF. Udowodniono, że jakakolwiek rozproszona implementacja LF wymaga co najmniej D rund, gdzie D jest maksymalnym stopniem wierzchołka w grafie.
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Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees
PublicationAn edge coloring of a graph G is called interval edge coloring if for each v ∈ V(G) the set of colors on edges incident to v forms an interval of integers. A graph G is interval colorable if there is an interval coloring of G. For an interval colorable graph G, by the interval chromatic index of G, denoted by χ'_i(G), we mean the smallest number k such that G is interval colorable with k colors. A bipartite graph G is called (α,β)-biregular...
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Parallel tabu search for graph coloring problem
PublicationTabu 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 patallel tabu search algorithms and compares three parallelization techniques applied to Tabucol, a sequential TS algorithm for graph coloring. The experimental results are based on graphs available from the DIMACS...
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A better practical algorithm for distributed graph coloring
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Interval vertex-coloring of a graph with forbidden colors
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Interval Vertex-Coloring of a Graph With Forbidden Colors
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Interval edge coloring of a graph with forbidden colors
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Distributed largest-first algorithm for graph coloring.
PublicationW artykule zaprezentowano rozproszony, probabilistyczny algorytm kolorowania grafów. Kolorowanie uzyskane jest optymalne lub prawie optymalne dla takich klas grafów jak koła dwudzielne, gąsienice czy korony. Udowodniono, że algorytm ten działa w czasie O(D^2 log n) rund dla dowolnego grafu n wierzchołkowegoo stopniu maksymalnym D.
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An experimental study of distributed algorithms for graph coloring.
PublicationW pracy podano algorytm rozproszonego kolorowania grafówi porównano ze znanym wcześniej algorytmem.
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A note on polynomial algorithm for cost coloring of bipartite graphs with Δ ≤ 4
PublicationIn the note we consider vertex coloring of a graph in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of coloring is the sum of costs incurred at each vertex. We show that the minimum cost coloring problem for n-vertex bipartite graph of degree ∆≤4 can be solved in O(n^2) time. This extends Jansen’s result [K.Jansen,The optimum cost chromatic partition problem, in:...
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On the complexity of distributed graph coloring with local minimality constraints
PublicationArtykuł traktuje o zachłannym kolorowaniu grafów w modelu rozproszonym. Omówiono algorytmy rozproszone, dające w wyniku pokolorowanie spełniające warunki dla pokolorowań sekwencyjnych typu S oraz Largest-First (LF). Udowodniono również, że każda rozproszona implementacja algorytmu S wymaga co najmniej Omega(log n / log log n) rund, a algorytmu LF co najmniej Omega (n^{1/2}) rund, gdzie n oznacza liczbę wierzchołków grafu.
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Greedy algorithms for backbone graph coloring in KOALA library
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Self-stabilizing algorithms for graph coloring with improved performance guarantees
PublicationW pracy rozważa się rozproszony model obliczeń, w którym struktura systemu jest reprezentowana przez graf bezpośrednich połączeń komunikacyjnych. W tym modelu podajemy nowy samostabilizujący algorytm kolorowania grafów oparty na konstrukcji drzewa spinającego. Zgodnie z naszą wiedzą jest to pierwszy algorytm z gwarantowaną wielomianową liczbą ruchów, który dokładnie koloruje grafy dwudzielne.
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Computer experiments with a parallel clonal selection algorithm for the graph coloring problem
PublicationArtificial immune systems (AIS) are algorithms that are based on the structure and mechanisms of the vertebrate immune system. Clonal selection is a process that allows lymphocytes to launch a quick response to known pathogens and to adapt to new, previously unencountered ones. This paper presents a parallel island model algorithm based on the clonal selection principles for solving the Graph Coloring Problem. The performance of...
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Scheduling with precedence constraints: mixed graph coloring in series-parallel graphs.
PublicationW pracy rozważono problem kolorowania grafów mieszanych, opisujący zagadnienie szeregowania zadań, w którym zależności czasowe zadań mają charakter częściowego porządku lub wzajemnego wykluczania. Dla przypadku, w którym graf zależności jest szeregowo-równoległy, podano algorytm rozwiązujący problem optymalnie w czasie $O(n^3.376 * log n)$.
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A graph coloring approach to scheduling of multiprocessor tasks on dedicated machines with availability constraints
PublicationWe address a generalization of the classical 1- and 2-processor unit execution time scheduling problem on dedicated machines. In our chromatic model of scheduling machines have non-simultaneous availability times and tasks have arbitrary release times and due dates. Also, the versatility of our approach makes it possible to generalize all known classical criteria of optimality. Under these stipulations we show that the problem...
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Non-disjoint Decomposition Using r-admissibility and Graph Coloring and Its Application in Index Generation Functions Minimization
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Kolorowanie grafów z ograniczeniami na liczbę wierzchołków w określonym kolorze = Graph coloring model with restrictions on cardinalities of vertexes in particular color
PublicationW artykule rozważamy problem takiego kolorowania grafów, w którym klasy kolorów mają ograniczoną z góry moc. Zagadnie to znajduje ciekawe zastosowania praktyczne i jest naturalnym uogólnieniem problemu kolorowania grafów. W artykule ustalamy złożoność obliczeniową dla grafów pełnych $r$-dzielnych i dla kilku innych prostych klas grafów oraz dla problemu dwukolorowania.
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Linear game non-contextuality and Bell inequalities—a graph-theoretic approach
PublicationWe study the classical and quantum values of a class of one-and two-party unique games, that generalizes the well-known XOR games to the case of non-binary outcomes. In the bipartite case the generalized XOR(XOR-d) games we study are a subclass of the well-known linear games. We introduce a 'constraint graph' associated to such a game, with the constraints defining the game represented by an edge-coloring of the graph. We use the...
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Edge coloring of graphs of signed class 1 and 2
PublicationRecently, Behr (2020) introduced a notion of the chromatic index of signed graphs and proved that for every signed graph (G, σ) it holds that ∆(G) ≤ χ′(G,σ) ≤ ∆(G) + 1, where ∆(G) is the maximum degree of G and χ′ denotes its chromatic index. In general, the chromatic index of (G, σ) depends on both the underlying graph G and the signature σ. In the paper we study graphs G for which χ′(G, σ) does not depend on σ. To this aim we...
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Equitable coloring of corona multiproducts of graphs
PublicationWe give some results regarding the equitable chromatic number for l-corona product of two graphs: G and H, where G is an equitably 3- or 4-colorable graph and H is an r-partite graph, a cycle or a complete graph. Our proofs lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of G.
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Interval incidence coloring of subcubic graphs
PublicationIn this paper we study the problem of interval incidence coloring of subcubic graphs. In [14] the authors proved that the interval incidence 4-coloring problem is polynomially solvable and the interval incidence 5-coloring problem is N P-complete, and they asked if χii(G) ≤ 2∆(G) holds for an arbitrary graph G. In this paper, we prove that an interval incidence 6-coloring always exists for any subcubic graph G with ∆(G) = 3.
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Dynamic F-free Coloring of Graphs
PublicationA problem of graph F-free coloring consists in partitioning the vertex set of a graph such that none of the resulting sets induces a graph containing a fixed graph F as an induced subgraph. In this paper we consider dynamic F-free coloring in which, similarly as in online coloring, the graph to be colored is not known in advance; it is gradually revealed to the coloring algorithm that has to color each vertex upon request as well...
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Rearrangeability in multicast Clos networks is NP-complete
PublicationPrzestrajalność w polach Closa z połączeniami jeden do jeden jest problemem wielomianowym. W pracy pokazano, że w polach z połączeniami jeden do wiele problem ten jest NP zupełny.Three-stage elos networks are commutation networks with circuit switching. So far, graph theory has been very useful tool for solving issues related to these networks with unicast connections. This is so because if elos network is represented as a bipartite...
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Sharp bounds for the complexity of semi-equitable coloring of cubic and subcubic graphs
PublicationIn this paper we consider the complexity of semi-equitable k-coloring of the vertices of a cubic or subcubic graph. We show that, given n-vertex subcubic graph G, a semi-equitable k-coloring of G is NP-hard if s >= 7n/20 and polynomially solvable if s <= 7n/21, where s is the size of maximum color class of the coloring.
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On-line P-coloring of graphs
PublicationFor 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...
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Equitable and semi-equitable coloring of cubic graphs and its application in batch scheduling
PublicationIn the paper we consider the problems of equitable and semi-equitable coloring of vertices of cubic graphs. We show that in contrast to the equitable coloring, which is easy, the problem of semi-equitable coloring is NP- complete within a broad spectrum of graph parameters. This affects the complexity of batch scheduling of unit-length jobs with cubic incompatibility graph on three uniform processors to minimize...
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The Backbone Coloring Problem for Small Graphs
PublicationIn this paper we investigate the values of the backbone chromatic number, derived from a mathematical model for the problem of minimization of bandwidth in radio networks, for small connected graphs and connected backbones (up to 7 vertices). We study the relationship of this parameter with the structure of the graph and compare the results with the solutions obtained using the classical graph coloring algorithms (LF, IS), modified...
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2-Coloring number revisited
Publication2-Coloring number is a parameter, which is often used in the literature to bound the game chromatic number and other related parameters. However, this parameter has not been precisely studied before. In this paper we aim to fill this gap. In particular we show that the approximation of the game chromatic number by the 2-coloring number can be very poor for many graphs. Additionally we prove that the 2-coloring number may grow...
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Dynamic coloring of graphs
PublicationDynamics 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 in WDM optical networks as well as traffic scheduling. In...
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Optimal backbone coloring of split graphs with matching backbones
PublicationFor a graph G with a given subgraph H, the backbone coloring is defined as the mapping c: V(G) -> N+ such that |c(u)-c(v)| >= 2 for each edge uv \in E(H) and |c(u)-c(v)| >= 1 for each edge uv \in E(G). The backbone chromatic number BBC(G;H) is the smallest integer k such that there exists a backbone coloring with max c(V(G)) = k. In this paper, we present the algorithm for the backbone coloring of split graphs with matching backbone.
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Tight bounds on the complexity of semi-equitable coloring of cubic and subcubic graphs
PublicationWe consider the complexity of semi-equitable k-coloring, k>3, of the vertices of a cubic or subcubic graph G. In particular, we show that, given a n-vertex subcubic graph G, it is NP-complete to obtain a semi-equitable k-coloring of G whose non-equitable color class is of size s if s>n/3, and it is polynomially solvable if s, n/3.
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Eqiuitable coloring of corona products of cubic graphs is harder than ordinary coloring
PublicationA graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G. In this paper the problem of determinig the equitable coloring number for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas...
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Minimum order of graphs with given coloring parameters
PublicationA complete k-coloring of a graph G=(V,E) is an assignment F: V -> {1,...,k} of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in a complete coloring (chromatic number χ(G) and achromatic number ψ(G), respectively),...
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Equitable coloring of corona products of graphs
PublicationIn this paper we consider an equitable coloring of some corona products of graphs G and H in symbols, G o H). In particular, we show that deciding the colorability of G o H is NP-complete even if G is 4-regular and H is K_2. Next, we prove exact values or upper bounds on the equitable chromatic number of G o H, where G is an equitably 3- or 4-colorable graph and H is an r-partite graph, a path, a cycle or a complete graph.
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The computational complexity of the backbone coloring problem for bounded-degree graphs with connected backbones
PublicationGiven a graph G, a spanning subgraph H of G and an integer λ>=2, a λ-backbone coloring of G with backbone H is a vertex coloring of G using colors 1, 2, ..., in which the color difference between vertices adjacent in H is greater than or equal to lambda. The backbone coloring problem is to find such a coloring with maximum color that does not exceed a given limit k. In this paper, we study the backbone coloring problem for bounded-degree...
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On some Zarankiewicz numbers and bipartite Ramsey Numbers for Quadrilateral
PublicationThe Zarankiewicz number z ( m, n ; s, t ) is the maximum number of edges in a subgraph of K m,n that does not contain K s,t as a subgraph. The bipartite Ramsey number b ( n 1 , · · · , n k ) is the least positive integer b such that any coloring of the edges of K b,b with k colors will result in a monochromatic copy of K n i ,n i in the i -th color, for some i , 1 ≤ i ≤ k . If n i = m for all i , then we denote this number by b k ( m )....
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Scheduling of identical jobs with bipartite incompatibility graphs on uniform machines. Computational experiments
PublicationWe consider the problem of scheduling unit-length jobs on three or four uniform parallel machines to minimize the schedule length or total completion time. We assume that the jobs are subject to some types of mutual exclusion constraints, modeled by a bipartite graph of a bounded degree. The edges of the graph correspond to the pairs of jobs that cannot be processed on the same machine. Although the problem is generally NP-hard,...
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On Computational Aspects of Greedy Partitioning of Graphs
PublicationIn this paper we consider a problem of graph P-coloring consisting in partitioning the vertex set of a graph such that each of the resulting sets induces a graph in a given additive, hereditary class of graphs P. We focus on partitions generated by the greedy algorithm. In particular, we show that given a graph G and an integer k deciding if the greedy algorithm outputs a P-coloring with a least k colors is NP-complete for an infinite...
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Computational aspects of greedy partitioning of graphs
PublicationIn this paper we consider a variant of graph partitioning consisting in partitioning the vertex set of a graph into the minimum number of sets such that each of them induces a graph in hereditary class of graphs P (the problem is also known as P-coloring). We focus on the computational complexity of several problems related to greedy partitioning. In particular, we show that given a graph G and an integer k deciding if the greedy...