Improper Coloring of Weighted Grid and Hexagonal Graphs

Bermond, Jean‐Claude; Havet, Frédéric; Huc, Florian; Sales, Cláudia Linhares

doi:10.1142/s1793830910000747

Cited by 7 publications

(9 citation statements)

References 9 publications

(10 reference statements)

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“…Lately, the problem has attracted renewed interest due to its applicability to communication networks, with the coloring of the graph modeling the assignment of frequencies to nodes and ∆ * representing some amount of tolerable interference. This has led to the study of the problem on Unit Disk Graphs [24] as well as various classes of grids [3,7,5]. Weighted generalizations have also been considered [6,23].…”

Section: Introductionmentioning

confidence: 99%

Defective Coloring on Classes of Perfect Graphs

Belmonte

Lampis

Mitsou

2017

Graph-Theoretic Concepts in Computer Science

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Abstract. In Defective Coloring we are given a graph G and two integers χ d , ∆ * and are asked if we can χ d -color G so that the maximum degree induced by any color class is at most ∆ * . We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters χ d , ∆ * is set to the smallest possible fixed value that does not trivialize the problem (χ d = 2 or ∆ * = 1). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either χ d or ∆ * is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both χ d and ∆ * are unbounded.

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Section: Introductionmentioning

confidence: 99%

Defective Coloring on Classes of Perfect Graphs

Belmonte

Lampis

Mitsou

2017

Graph-Theoretic Concepts in Computer Science

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“…Since this subject is too wide to be surveyed in a short paper, we mention just a few examples like subcoloring known also as P 3 -free coloring (where P p denotes the chordless path on p vertices), P 4 -free coloring and improper coloring, and we refer to appropriate literature on other variants, e.g., many results on subcoloring can be found in Albertson et al [2], Broere and Mynhardt [8], Fiala et al [16] as well as in work of Gimbel and Hartman [17]. For results on P 4 -free coloring see, e.g., Gimbel and Nešetřil [18] and a paper of Hoàng and Le [23], while for improper coloring we refer the reader to papers of Bermond et al [4], Cowen et al [15] and Havet et al [21]. Concerning the computational complexity of F-free k-coloring problem we mention the result of Achlioptas [1] who proved that for any fixed graph F, except K 2 , the problem of deciding if a given graph admits an F-free coloring with at most k-colors is NP-complete (for a detailed study of the computational complexity of many variants of offline generalized colorings see, e.g., Broersma et al [9]).…”

Section: Related Research and Our Resultsmentioning

confidence: 99%

Dynamic F-free Coloring of Graphs

Borowiecki

Sidorowicz

2018

Graphs and Combinatorics

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A 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 as handle any vertex recoloring requests. Our main concern is the greedy approach and characterization of graph classes for which it is possible to decide in polynomial time whether for the fixed forbidden graph F and positive integer k the greedy algorithm ever uses more than k colors in dynamic F-free coloring. For various classes of graphs we give such characterizations in terms of a finite number of minimal forbidden graphs thus solving the above-mentioned problem for the so-called F-trees when F is 2-connected, and for classical trees, when F is a path of order 3 (the latter variant is also known as subcoloring or 1-improper coloring).

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“…Again, the algorithm has a running time of at most O(n 3 ). 4 Upper bounds on f (k, d) and determination of f (k, d) for…”

Section: Output: Smentioning

confidence: 99%

New Approach to the $k$-Independence Number of a Graph

Caro¹,

Hansberg²

2013

Electron. J. Combin.

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Let G = (V, E) be a graph and k ≥ 0 an integer. A k-independent set S ⊆ V is a set of vertices such that the maximum degree in the graph induced by S is at most k. With α k (G) we denote the maximum cardinality of a k-independent set of G. We prove that, for a graph G on n vertices and average degree d, α k (G) ≥ k+1 ⌈d⌉+k+1 n, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on k-independence, J. Graph Theory 15 (1991), 99-107].

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Improper Coloring of Weighted Grid and Hexagonal Graphs

Cited by 7 publications

References 9 publications

Defective Coloring on Classes of Perfect Graphs

Defective Coloring on Classes of Perfect Graphs

Dynamic F-free Coloring of Graphs

New Approach to the $k$-Independence Number of a Graph

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