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Search results for: GRAPH COLORING
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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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The complexity of the T-coloring problem for graphs with small degree.
PublicationW pracy ustalono złożoność obliczeniową problemu optymalnego kolorowania grafów o ustalonym stopniu.
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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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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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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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Some results concerning the complexity of restricted colorings of graphs
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A note on compact and compact circular edge-colorings of graphs
PublicationW pracy rozważamy dwa warianty kolorowania krawędzi grafów prostych i ważonych, mianowicie kolorowania zwarte oraz zwarte cyrkularne. Rozważamy relacje pomiędzy nimi. Dowodzimy, że każdy zewnętrznie planarny graf dwudzielny posiada zwarte pokolorowanie krawędziowe oraz, że problem ten dla grafów ogólnych jest NP-zupełny. Podajemy również wielomianowy 1.5-przybliżony algorytm oraz pseudowielomianowy dokładny algorytm zwartego cyrkularnego...
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Efficient list cost coloring of vertices and/or edges of bounded cyclicity graphs
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Efficient List Cost Coloring of Vertices and∕or Edges of Some Sparse Graphs
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Equitable Coloring of Graphs. Recent Theoretical Results and New Practical Algorithms
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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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Efficient list cost coloring of vertices and/or edges of bounded cyclicity graphs
PublicationW artykule rozważamy listowo-kosztowe kolorowanie wierzchołków i krawędzi grafu w modelu wierzchołkowym, krawędziowym, totalnym i pseudototalnym. Stosujemy programowanie dynamiczne w celu otrzymania algorytmów wielomianowych dla drzew. Następnie uogólniamy to podejście na dowolne grafy z ograniczonymi liczbami cyklomatycznymi i na ich multikolorowania.
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The computational complexity of the backbone coloring problem for planar graphs with connected backbones
PublicationIn the paper we study the computational complexity of the backbone coloring problem for planar graphs with connected backbones. For every possible value of integer parameters λ≥2 and k≥1 we show that the following problem: Instance: A simple planar graph GG, its connected spanning subgraph (backbone) HH. Question: Is there a λ-backbone coloring c of G with backbone H such that maxc(V(G))≤k? is either NP-complete or polynomially...
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On Optimal Backbone Coloring of Split and Threshold Graphs with Pairwise Disjoint Stars
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Efficient list cost coloring of vertices and/or edges of some sparse graphs
PublicationRozważane jest kolorowanie wierzchołków i krawędzi grafów w modelach klasycznym, totalnym i pseudototalnym z uwzględnieniem dodatkowego ograniczenia w postaci list dostępnych kolorów. Proponujemy wielomianowy algorytm oparty na paradygmacie programowania dynamicznego dla grafów o strukturze drzewa. Wynik ten można uogólnić na grafy o liczbie cyklomatycznej ograniczonej z góry przez dowolnie wybraną stała.
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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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Equitable coloring of graphs. Recent theoretical results and new practical algorithms
PublicationIn this paper we survey recent theoretical results concerning conditions for equitable colorability of some graphs and recent theoretical results concerning the complexity of equitable coloring problem. Next, since the general coloring problem is strongly NP-hard, we report on practical experiments with some efficient polynomial-time algorithms for approximate equitable coloring of general graphs.
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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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Equitable colorings of some variation of corona products of cubic graphs
PublicationThe problem of determining the value of equitable chromatic number for multicoronas of cubic graphs is studied. We provide some polynomially solvable cases of cubical multicoronas and give simple linear time algorithms for equitable coloring of such graphs which use almost optimal number of colors in the remaining cases.
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Equitable and semi-equitable coloring of cubic graphs and its application in batch scheduling
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A note on fast approximate backbone coloring of split graphs with star--like backbones
PublicationDla grafu G = (V, E) z wyróżnionym podgrafem H, kolorowanie szkieletowe jest zdefiniowane jako odwzorowanie c spełniające |c(u) - c(v)| > 1 dla każdej krawędzi z E(H) oraz |c(u) - c(v)| > 0 dla każdej krawędzi z E(G). W pracy przedstawiono 1-przybliżony algorytm kolorowania szkieletowego split grafów ze skojarzeniem w szkielecie o złożoności O(|V|) oraz 1-przybliżony algorytm dla split grafów z rozłącznymi gwiazdami w szkielecie.
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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 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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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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Equitable coloring of hypergraphs
PublicationA hypergraph is equitablyk-colorable if its vertices can be partitioned into k sets/colorclasses in such a way that monochromatic edges are avoided and the number of verticesin any two color classes differs by at most one. We prove that the problem of equitable 2-coloring of hypergraphs is NP-complete even for 3-uniform hyperstars. Finally, we apply the method of dynamic programming for designing a polynomial-time algorithm to...
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On the complexity of distributed greedy coloring
PublicationW pracy rozważono problem kolorowania grafów przy dodatkowym założeniu, że kolor żadnego wierzchołka nie może zostać zmniejszony bez zmiany kolorów przynajmniej jednego z jego sąsiadów. Przeprowadzone rozważania dotyczyły złożoności obiczeniowej problemu w modelu Liniala obliczeń rozproszonych. Podano ograniczenia dolne i górne złożoności problemu oraz zestawiono problem z innymi pokrewnymi zagadnieniami grafowymi.
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A note on mixed tree coloring
PublicationZaproponowano liniowy algorytm dla problemu kolorowania mieszanego w drzewach, uzyskując tym samym poprawę w stosunku do algorytmu o złożoności O(n^2) podanego w pracy [P. Hansen, J. Kuplinsky, D. de Werra, Mixed graph colorings, Math. Methods Oper. Res. 45 (1997) 145-160].
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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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EvOLAP Graph – Evolution and OLAP-Aware Graph Data Model
PublicationThe objective of this paper is to propose a graph model that would be suitable for providing OLAP features on graph databases. The included features allow for a multidimensional and multilevel view on data and support analytical queries on operational and historical graph data. In contrast to many existing approaches tailored for static graphs, the paper addresses the issue for the changing graph schema. The model, named Evolution...
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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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Edge-coloring of 3-uniform hypergraphs
PublicationWe consider edge-colorings of 3-uniform hypergraphs which is a natural generalization of the problem of edge-colorings of graphs. Various classes of hypergraphs are discussed and we make some initial steps to establish the border between polynomial and NP-complete cases. Unfortunately, the problem appears to be computationally difficult even for relatively simple classes of hypergraphs.
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On the hat problem on a graph
PublicationThe topic of this paper is the hat problem in which each of n players is uniformly and independently fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of winning....
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Hat problem on a graph
PublicationThe topic of our paper is the hat problem. In that problem, each of n people is randomly fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color looking at the hat colors of the other people. The team wins if at least one person guesses his hat color correctly and no one guesses his hat color wrong, otherwise the team loses. The aim is to maximize the probability of win. In this version every...
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Graph security testing
PublicationSet S ⊂ V is called secure set iff ∀ X ⊂ S | N [ X ] ∩ S | ≥ | N ( X ) \ S | [3]. That means that every subset of a secure set has at least as many friends (neighbour vertices in S) as enemies (neighbour vertices outside S) and will be defended in case of attack. Problem of determining if given set is secure is co −NP -complete, there is no efficient algorithm solving it [3]. Property testers are algorithms that distinguish inputs...
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Restrained differential of a graph
PublicationGiven a graph $G=(V(G), E(G))$ and a vertex $v\in V(G)$, the {open neighbourhood} of $v$ is defined to be $N(v)=\{u\in V(G) :\, uv\in E(G)\}$. The {external neighbourhood} of a set $S\subseteq V(G)$ is defined as $S_e=\left(\cup_{v\in S}N(v)\right)\setminus S$, while the \emph{restrained external neighbourhood} of $S$ is defined as $S_r=\{v\in S_e : N(v)\cap S_e\neq \varnothing\}$. The restrained differential of a graph $G$ is...
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New potential functions for greedy independence and coloring
PublicationA potential function $f_G$ of a finite, simple and undirected graph $G=(V,E)$ is an arbitrary function $f_G : V(G) \rightarrow \mathbb{N}_0$ that assigns a nonnegative integer to every vertex of a graph $G$. In this paper we define the iterative process of computing the step potential function $q_G$ such that $q_G(v)\leq d_G(v)$ for all $v\in V(G)$. We use this function in the development of new Caro-Wei-type and Brooks-type...
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Optimal edge-coloring with edge rate constraints
PublicationWe consider the problem of covering the edges of a graph by a sequence of matchings subject to the constraint that each edge e appears in at least a given fraction r(e) of the matchings. Although it can be determined in polynomial time whether such a sequence of matchings exists or not [Grötschel et al., Combinatorica (1981), 169–197], we show that several questions about the length of the sequence are computationally intractable....
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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...
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Forwarding and optical indices of a graph
PublicationW pracy rozstrzygnięto dwa problemy dotyczące komunikacji wszyscy-do-wszystkich w grafach. Stwierdzono, że dla wersji skierowanej problemu parametry ''pi'' (maksymalne obciążenie krawędzi) i ''w'' (parametr chromatyczny) nie muszą być w ogólności sobie równe. Dla wersji nieskierowanej problemu pokazano, że wyznaczenie wartości zarówno ''pi'', jak i ''w'', jest w ogólności problemem NP-trudnym.
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Zgred - Zgred Graph Editor
PublicationThe paper presents a graph editor that was developed as a supplementary tool for the Koala graph library. We discuss its requirements, design choices and main ideas behind the implementation. The later part of the paper is meant to be a brief user manual, as we go through the functionality provided by the editor
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Fast Collaborative Graph Exploration
PublicationWe study the following scenario of online graph exploration. A team of k agents is initially located at a distinguished vertex r of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure...
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Fast collaborative graph exploration
PublicationWe study the following scenario of online graph exploration. A team of k agents is initially located at a distinguished vertex r of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure...
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On a Recurrence Arising in Graph Compression
PublicationIn a recently proposed graphical compression algorithm by Choi and Szpankowski (2012), the following tree arose in the course of the analysis. The root contains n balls that are consequently distributed between two subtrees according to a simple rule: In each step, all balls independently move down to the left subtree (say with probability p) or the right subtree (with probability 1p). A new node is created as long as...
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Graph models of clos networks
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Graph classes generated by Mycielskians
PublicationIn this paper we use the classical notion of weak Mycielskian M'(G) of a graph G and the following sequence: M'_{0}(G) =G, M'_{1}(G)=M'(G), and M'_{n}(G)=M'(M'_{n−1}(G)), to show that if G is a complete graph oforder p, then the above sequence is a generator of the class of p-colorable graphs. Similarly, using Mycielskian M(G) we show that analogously defined sequence is a generator of the class consisting of graphs for which the...
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Parallel scheduling by graph ranking
PublicationNr dokum.: 73017Praca dotyczy jednego z nieklasycznych modeli kolorowania grafów - uporządkowanego kolorowania. Celem było uzyskanie wyników, które mogo być wykorzystane w praktycznych zastosowaniach tego modelu, do których należą: równoległe przetwarzanie zapytań w relacyjnych bazach danych, równoległa faktoryzacja macierzy metodą Choleskiego, równoległa asemblacja produktu z jego części składowych. W pracy wskazano uogólnienia...
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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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Sprawiedliwe i półsprawiedliwe pokolorowania grafów kubicznych
PublicationW pracy rozpatrywane są sprawiedliwe i półsprawiedliwe pokolorowania grafów kubicznych. Pokazano, że w odróżnieniu od tego pierwszego, który jest łatwy, problem istnienia pokolorowań półsprawiedliwych jest NP-zupełny w szerokim zakresie parametrów grafów.
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Some results on a trading model in a consensus list coloring
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