Coloring the Maximal Cliques of Graphs

Bacsó, Gábor; Gravier, Sylvain; Gyárfás, András; Preissmann, Myriam; Sebő, András

doi:10.1137/s0895480199359995

Cited by 66 publications

(90 citation statements)

References 13 publications

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“…Another interesting problem is the problem of Clique coloring, which is a variation of the classical vertex coloring graph coloring. Clique coloring requires that only every inclusion-wise maximal (not extendable) clique contains at least two different colors [2], instead of requiring that the two end points of each edge have two different colors.…”

Section: Introductionmentioning

confidence: 99%

“…Finding the minimum number of colors for a given graph or finding the maximum clique in a graph is considered NP-complete, i.e., they run in a nonpolynomial time. Detecting the maximum clique of a graph can be very useful for finding a minimal coloring for a graph, where a k-clique-colorable graph has a k chromatic number [2]. Although, this may not hold for all graphs, e.g., 2-clique and 3-clique-colorable graphs, the k-clique-colorable can be a reasonable bound for the chromatic number of a graph [2].…”

Section: Introductionmentioning

confidence: 99%

“…Detecting the maximum clique of a graph can be very useful for finding a minimal coloring for a graph, where a k-clique-colorable graph has a k chromatic number [2]. Although, this may not hold for all graphs, e.g., 2-clique and 3-clique-colorable graphs, the k-clique-colorable can be a reasonable bound for the chromatic number of a graph [2]. Hence, if a coloring heuristic algorithm manages to first color the nodes in a clique with k colors (k-clique-colorable) before moving to other nodes, then the algorithm is more likely to find a minimal coloring of the graph.…”

Section: Introductionmentioning

confidence: 99%

“…This means that there is no known algorithm for solving this problem in a polynomial time, except for those which have been developed for specialized graphs such as the planer graphs or perfect graphs where the problem can be solved in a polynomial time [2,6,7,8,14,15]. Solving this problem can also be done in polynomial time if k is constant, where k is the number of nodes in the clique; in this case all subgraphs of at least k nodes will be checked whether they form a clique or not.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Maximum Clique Conformance Measure for Graph Coloring Algorithms

Alzubi¹,

Hassan²,

Malkawi³

2015

IJCA

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The Graph Coloring Problem (GCP) is an essential problem in graph theory, and it has many applications such as the exam scheduling problem, register allocation problem, timetabling, and frequency assignment. The maximum clique problem is also another important problem in graph theory and it arises in many real life applications like managing the social networks. These two problems have received a lot of attention by the scientists, not only for their importance in the real life applications, but also for their theoretical aspect. Solving these problems however remains a challenge, and the proposed algorithms for this purpose apply to rather small graphs, whereas many real life application graphs encompass hundreds or thousands of nodes. This paper presents a new measure for evaluating the efficiency of graph coloring algorithms. The new measure computes the clique conformance index (CCI) of a graph coloring algorithm. CCI measures the rate of deviation of a coloring algorithm from the maximum clique during the process of coloring a graph. The paper presents empirical measurement for two coloring algorithms proposed by the authors.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Maximum Clique Conformance Measure for Graph Coloring Algorithms

Alzubi¹,

Hassan²,

Malkawi³

2015

IJCA

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show abstract

“…A consequence of this theorem is that properties established for generalized split graphs are immediately properties of almost all C 5 -free (and almost all perfect) graphs. Bacsó et al [5] employ this technique to show that the clique-hypergraphs of almost all perfect graphs are 3-colorable.…”

Section: Introductionmentioning

confidence: 99%

Algorithms for unipolar and generalized split graphs

Eschen

Wang

2014

Discrete Applied Mathematics

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A graph G = (V, E) is a unipolar graph if there exists a partition V = V 1 ∪ V 2 such that, V 1 is a clique and V 2 induces the disjoint union of cliques. The complement-closed class of generalized split graphs contains those graphs G such that either G or the complement of G is unipolar. Generalized split graphs are a large subclass of perfect graphs. In fact, it has been shown that almost all C 5 -free (and hence, almost all perfect graphs) are generalized split graphs. In this paper we present a recognition algorithm for unipolar graphs that utilizes a minimal triangulation of the given graph, and produces a partition when one exists. Our algorithm has running time O(nm + nm F ), where m F is the number of edges added in a minimal triangulation of the given graph. Generalized split graphs can be recognized via this algorithm in O(n 3 ) time. We give algorithms on unipolar graphs for finding a maximum independent set and a minimum clique cover in O(n + m) time and for finding a maximum clique and a minimum proper coloring in O(n 2.5 / log n) time, when a unipolar partition is given. These algorithms yield algorithms for the four optimization problems on generalized split graphs that have the same worst-case time bounds. We also report that the perfect code problem is NP-Complete for chordal unipolar graphs.

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Clique-coloring some classes of odd-hole-free graphs

Défossez¹

2006

J. Graph Theory

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Abstract:We consider the problem of clique-coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd-hole-free graph that is not 3-clique-colorable, the existence of a constant C such that any perfect graph is C-clique-colorable is an open problem. In this paper we solve this problem for some subclasses of odd-hole-free graphs: those that are diamond-free and those that are bull-free. We also prove the NP-completeness of 2-cliquecoloring K 4 -free perfect graphs.

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Coloring the Maximal Cliques of Graphs

Cited by 66 publications

References 13 publications

Maximum Clique Conformance Measure for Graph Coloring Algorithms

Maximum Clique Conformance Measure for Graph Coloring Algorithms

Algorithms for unipolar and generalized split graphs

Clique-coloring some classes of odd-hole-free graphs

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