Topic
Equitable coloring
About: Equitable coloring is a research topic. Over the lifetime, 168 publications have been published within this topic receiving 2206 citations.
Papers published on a yearly basis
Papers
14 Aug 2007
TL;DR: The problem of determining whether χl(G) ≤ r, the LIST CHROMATIC NUMBER problem, is solvable in linear time for every fixed treewidth bound t, and a list-based variation, LIST EQUITABLE COLORING is W[1]-hard for trees, parameterized by the number of colors on the lists.
Abstract: We study the complexity of several coloring problems on graphs, parameterized by the treewidth t of the graph: (1) The list chromatic number χl(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 Lv of colors, where each list has length at least r, there is a choice of one color from each vertex list Lv yielding a proper coloring of G. We show that the problem of determining whether χl(G) ≤ r, the LIST CHROMATIC NUMBER problem, is solvable in linear time for every fixed treewidth bound t. The method by which this is shown is new and of general applicability. (2) The LIST COLORING problem takes as input a graph G, together with an assignment to each vertex v of a set of colors Cv. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors Cv, 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. (3) 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 (t, r). We also show that a list-based variation, LIST EQUITABLE COLORING is W[1]-hard for trees, parameterized by the number of colors on the lists.
129 citations
1 Jan 1998
TL;DR: A survey of recent progress on the equitable coloring of graphs can be found in this article, where the authors pay more attention to work done on the Equitable ∆-Coloring Conjecture.
Abstract: Let the vertices of a graph G be colored with k colors such that no adjacent vertices receive the same color and the sizes of the color classes differ by at most one. Then G is said to be equitably k-colorable. The equitable chromatic number x = (G) is the smallest integer k such that G is equitably k-colorable. In this article, we survey recent progress on the equitable coloring of graphs. We pay more attention to work done on the Equitable ∆-Coloring Conjecture. We also discuss related graph coloring notions and their problems. The survey ends with suggestions for further research topics.
103 citations
1 Jan 2013
TL;DR: The equitable chromatic number as discussed by the authors is the smallest integer k such that a graph G is equitably k-colorable, i.e., no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one.
Abstract: If the vertices of a graph G are colored with k colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then G is said to be equitably k-colorable. The equitable chromatic number D .G/ is the smallest integer k such that G is equitably k-colorable. In the first introduction section, results obtained about the equitable chromatic number before 1990 are surveyed. The research on equitable coloring has attracted enough attention only since the early 1990s. In the subsequent sections, positive evidence for the important equitable -coloring conjecture is supplied from graph classes such as forests, split graphs, outerplanar graphs, series-parallel graphs, planar graphs, graphs with low degeneracy, graphs with bounded treewidth, Kneser graphs, and interval graphs. Then three kinds of graph products are investigated. A list version of equitable coloring is introduced. The equitable coloring is further examined in the wider context of graph packing. Appropriate conjectures for equitable -coloring of disconnected graphs are then studied. Variants of the well-known and significant Hajnal and Szemeredi Theorem are discussed. A brief summary of applications of equitable coloring is given. Related notions, such as equitable edge coloring, equitable total coloring, equitable defective coloring, and equitable coloring of uniform hypergraphs, are touched upon. This chapter ends with a short conclusion section. This survey is an updated version of Lih [102].
95 citations
TL;DR: It is shown that a non-null tree T as a bipartite graph T is equitably k -colorable if and only if its vertices can be partitioned into k independent sets of as near equal sizes as possible.
89 citations
TL;DR: In this article, the problem of determining whether a given graph G has an equitable k-coloring is solved in polynomial time on graphs of bounded treewidth, and a precolored version remains NP-complete on trees.
83 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 10 |
| 2020 | 11 |
| 2019 | 13 |
| 2018 | 17 |
| 2017 | 13 |
| 2016 | 12 |