Equitable colorings of bounded treewidth graphs

Bodlaender, Hans L.; Fomin, Fedor V.

doi:10.1016/j.tcs.2005.09.027

Cited by 57 publications

(40 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We give a proof that the running time n O(t) of algorithms from [4,20] for different variants of Graph Coloring is essentially the best we can hope for up to a widely believed assumption FPT = W [1]. In particular, we show that…”

Section: Our Resultsmentioning

confidence: 92%

“…The treewidth of G is bounded by k as the graph obtained after removing the vertices The Equitable Coloring problem is a classical problem with a long history starting from 1960s [19,25]. Bodlaender and Fomin have shown that determining whether a graph of treewidth at most t admits an equitable coloring, can be solved in time O(n O(t) ) [4].…”

Section: List Coloring and Precoloring Extension Are W [1]-hard Parammentioning

confidence: 99%

“…Due to the enforcements alluded to above, each "selection" coloring of a gadget (there will be only one possible in each group of gadgets) will force some number of vertices to be colored with these pairs of colors, which can be thought of as an information transmission. For example, when a gadget in G [4,2] is colored with a "selection" coloring, this indicates that the edge from which the gadget arises is selected as the edge from the color class V [4] of G, to the color class V [2]. There is a pair of colors that handles the information transmission concerning which edge is selected between the groups G [2,4] and G [4,2].…”

Section: The Numerical Targets (Function H)mentioning

confidence: 99%

See 2 more Smart Citations

On the complexity of some colorful problems parameterized by treewidth

Fellows

Fomin

Lokshtanov

et al. 2011

Information and Computation

Self Cite

116

View full text Add to dashboard Cite

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: In this paper, we study the complexity of several coloring problems on graphs, parameterized by the treewidth of the graph. 1. The List Coloring problem takes as input a graph G, together with an assignment to each vertex v of a set of colors C v. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors C v , for each vertex, so that the obtained coloring of G is proper. We show that this problem is W [1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W [1]-hard, parameterized by treewidth. 2. An equitable coloring of a graph G is a proper coloring of the vertices where the numbers of vertices having any two distinct colors differs by at most one. We show that the problem is hard for W [1], parameterized by the treewidth plus the number of colors. We also show that a list-based variation, List Equitable Coloring is W [1]-hard for forests, parameter-ized by the number of colors on the lists. 3. The list chromatic number χ l (G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list L v of colors, where each list has length at least r, there is a choice of one color from each vertex list L v yielding a proper coloring of G. We show that the problem of determining whether χ l (G) ≤ r, the List Chromatic Number problem, is solvable in linear time on graphs of constant treewidth.

show abstract

Section: Our Resultsmentioning

confidence: 92%

Section: List Coloring and Precoloring Extension Are W [1]-hard Parammentioning

confidence: 99%

Section: The Numerical Targets (Function H)mentioning

confidence: 99%

See 1 more Smart Citation

On the complexity of some colorful problems parameterized by treewidth

Fellows

Fomin

Lokshtanov

et al. 2011

Information and Computation

Self Cite

116

View full text Add to dashboard Cite

show abstract

“…For trees, both problems, equitable and b-bounded kcoloring, are studied in [1,5,6,7,14]; a linear-time algorithm that solves the b-bounded k-coloring problem is designed in the first reference. In [3] it is shown that the equitable k-coloring problem is polynomial-time solvable in bounded tree-width graphs but that the precolored version becomes N P -complete, even in trees.…”

Section: Introductionmentioning

confidence: 99%

Locally boundedk-colorings of trees

Bentz

Picouleau

2009

RAIRO-Oper. Res.

View full text Add to dashboard Cite

show abstract

“…N P-hard in the general case, the complexity of the mutual exclusion scheduling problem has been investigated for several classes of graphs, in particular some classes of perfect graphs (Baker and Coffman 1996;Bodlaender and Fomin 2004;Bodlaender and Jansen 1995;Cohen and Tarsi 1991;Dahlhaus and Karpinski 1998;De Werra 1997;Finke et al 2004;Hansen et al 1993;Jansen 2003;Jarvis and Zhou 2001;Lonc 1991). Unfortunately, few positive results have been published on the subject: for the majority of the classes studied, the problem is shown to be N P-hard.…”

Section: Introductionmentioning

confidence: 99%

Mutual exclusion scheduling with interval graphs or related classes: Complexity and algorithms

Gardi

2006

4OR

View full text Add to dashboard Cite

This note summarizes the main results presented in the author's Ph.D. thesis, supervised by Professor Michel Van Caneghem and defended on 14th June 2005 at University of Aix-Marseille II, France. The thesis, written in French, is available at http://www.lif-sud.univ-mrs.fr/ Rapports/25-2005.html. The mutual exclusion scheduling problem has an elegant graph-theoretic formulation: given an undirected graph G and an integer k, find a minimum coloring of G such that each color appears at most k times. When G is an interval graph, this problem has some applications in workforce planning. Then, the object of the thesis is to study the complexity of mutual exclusion scheduling problem for interval graphs and related classes.

show abstract

Equitable colorings of bounded treewidth graphs

Cited by 57 publications

References 24 publications

On the complexity of some colorful problems parameterized by treewidth

On the complexity of some colorful problems parameterized by treewidth

Locally boundedk-colorings of trees

Mutual exclusion scheduling with interval graphs or related classes: Complexity and algorithms

Contact Info

Product

Resources

About