A survey of graph coloring - its types, methods and applications

Formanowicz, Piotr; Tanaś, Krzysztof

doi:10.2478/v10209-011-0012-y

Cited by 42 publications

(19 citation statements)

References 27 publications

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“…Finding the edge coloring number (the minimum possible number of colors in an edge coloring) of a given graph is a classic problem of graph theory. The study of edge coloring is motivated by theoretical aspects of graph theory as well as by numerous applications in information theory and computer science (mostly by different types of scheduling problems, see a survey in [23]).…”

Section: Combinatorial Applications Of Theorem 1 a A Lower Boundmentioning

confidence: 99%

A Conditional Information Inequality and Its Combinatorial Applications

Kaced

Romashchenko

Vereshchagin

2018

IEEE Trans. Inform. Theory

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We show that the inequality H(A|B, X) + H(A|B, Y ) H(A|B) for jointly distributed random variables A, B, X, Y , which does not hold in general case, holds under some natural condition on the support of the probability distribution of A, B, X, Y . This result generalizes a version of the conditional Ingleton inequality: if for some distribution I(X :We present two applications of our result. The first one is the following easy-to-formulate theorem on edge colorings of bipartite graphs: assume that the edges of a bipartite graph are colored in K colors so that each two edges sharing a vertex have different colors and for each pair (left vertex x, right vertex y) there is at most one color a such both x and y are incident to edges with color a; assume further that the degree of each left vertex is at least L and the degree of each right vertex is at least R. Then K LR. The second application is a new method to prove lower bounds for biclique cover of bipartite graphs. KeywordsShannon entropy, conditional information inequalities, non Shannon type information inequalities, biclique cover, edge coloring

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Section: Combinatorial Applications Of Theorem 1 a A Lower Boundmentioning

confidence: 99%

A Conditional Information Inequality and Its Combinatorial Applications

Kaced

Romashchenko

Vereshchagin

2018

IEEE Trans. Inform. Theory

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“…The task is to assign a set of colors to each vertex, such that whenever two vertices are adjacent, their colors differ from each other. For a survey about this famous graph problem and related algorithms, see [1].…”

Section: Introductionmentioning

confidence: 99%

Computational complexity of distance edge labeling

Knop

Masařík

2018

Discrete Applied Mathematics

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Abstract. The problem of Distance Edge Labeling is a variant of Distance Vertex Labeling (also known as L2,1 labeling) that has been studied for more than twenty years and has many applications, such as frequency assignment. The Distance Edge Labeling problem asks whether the edges of a given graph can be labeled such that the labels of adjacent edges differ by at least two and the labels of edges at distance two differ by at least one. Labels are chosen from the set {0, 1, . . . , λ} for λ fixed. We present a full classification of its computational complexity-a dichotomy between the polynomially solvable cases and the remaining cases which are NP-complete. We characterise graphs with λ ≤ 4 which leads to a polynomial-time algorithm recognizing the class and we show NPcompleteness for λ ≥ 5 by several reductions from Monotone Not All Equal 3-SAT.

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“…Geetha and Somasundaram [12] proved the TCC for generalized Sierpiński graphs. A survey on graph coloring for its types, methods and applications are given in [13]. Recently [14] it is proved that the graphs K n × K n , C m × C n and G H are type-I graphs, where G is any bipartite graph.…”

Section: Introductionmentioning

confidence: 99%

Total Coloring Conjecture for Certain Classes of Graphs

Vignesh¹,

Geetha²,

Somasundaram³

2018

Algorithms

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A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ′ ′ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ′ ′ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.

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A survey of graph coloring - its types, methods and applications

Cited by 42 publications

References 27 publications

A Conditional Information Inequality and Its Combinatorial Applications

A Conditional Information Inequality and Its Combinatorial Applications

Computational complexity of distance edge labeling

Total Coloring Conjecture for Certain Classes of Graphs

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