# Vertex-Coloring with Defects

@article{Angelini2017VertexColoringWD,
title={Vertex-Coloring with Defects},
author={Patrizio Angelini and Michael A. Bekos and Felice De Luca and Walter Didimo and Michael Kaufmann and Stephen G. Kobourov and Fabrizio Montecchiani and Chrysanthi N. Raftopoulou and Vincenzo Roselli and Antonios Symvonis},
journal={J. Graph Algorithms Appl.},
year={2017},
volume={21},
pages={313-340}
}
• Published 2017
• Mathematics
• J. Graph Algorithms Appl.
Defective coloring is a variant of the traditional vertex-coloring in which adjacent vertices are allowed to have the same color, as long as the induced monochromatic components have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, diameter, and acyclicity of the monochromatic components. We focus on defective colorings with κ colors in which the…
13 Citations

## Figures from this paper

Bipartizing with a Matching
• Mathematics
COCOA
• 2018
This work studies the problem of determining whether a given graph G admits a matching M whose removal destroys all odd cycles of G and shows that this problem is fixed-parameter tractable when parameterized by the clique-width, which implies that it is polynomial-time solvable for many interesting graph classes, such as distance-hereditary, outerplanar, and chordal graphs.
Parameterized (Approximate) Defective Coloring
• Mathematics, Computer Science
STACS
• 2018
The complexity of Defective Coloring is investigated, and an ETH-based lower bound for treewidth and pathwidth is given, showing that no algorithm can solve the problem in $n^{o(pw)}$, essentially matching the complexity of an algorithm obtained with standard techniques.
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic
Complexity Analysis and Stochastic Convergence of Some Well-known Evolutionary Operators for Solving Graph Coloring Problem
• Mathematics, Computer Science
Mathematics
• 2020
The asymptotic analysis of some well-known and recent evolutionary operators for finding the chromatic number of a graph G and the necessary and sufficient conditions for the global convergence of evolutionary algorithms are focused on.
On the computational complexity of the bipartizing matching problem
• Mathematics
• 2017
The problem of determining whether a given graph~G=(V,E) admits a matching~$M$ whose removal destroys all odd cycles of~$G$ is studied, and it is shown that the problem is fixed parameter tractable when parameterized by the clique-width, which implies polynomial-time solution for many interesting graph classes, such as distance-hereditary, outerplanar, and chordal graphs.
A New Multi-Objective Evolutionary Approach to Graph Coloring and Channel Allocation Problems
• Computer Science
Journal of Applied Mathematics and Computation
• 2021
A new multi-objective evolutionary algorithm using the new evolutionary operators with multi- objectives in finding the solution to graph coloring and channel allocation and achieves better solution compared to some of the existing well known methods.
Solution to Graph Coloring Using Genetic and Tabu Search Procedures
• Mathematics
• 2018
Some of the engineering applications warrant the solution of Graph Coloring Problem, an NP-hard combinatorial optimization problem. This paper focuses on designing three new evolutionary operators
Graph Coloring In Optimization Total Waste Transport Vehicles In Bandung
• 2020
Waste is a material that is produced from human activities that are no longer used and disposed of. Waste management really needs support and commitment from the government to produce sustainable
Sobre (2,1)-colorações em grafos exoplanares maximais
• Anais do VI Encontro de Teoria da Computação (ETC 2021)
• 2021
Uma $(m, d)$-coloração $\pi$ de um grafo $G$ é uma atribuição de $m$ cores aos vértices do grafo de maneira que, para todo vértice $u$ de $G$, no máximo $d$ vizinhos de $u$ possuem a cor $\pi(u)$.

## References

SHOWING 1-10 OF 50 REFERENCES
Defective coloring revisited
• Mathematics
J. Graph Theory
• 1997
This paper investigates the existence of such colorings in surfaces and the complexity of coloring problems, and shows that the (2, k)-coloring, for k ≥ 1, and the (3, 1-coloring problems are NP-complete even for planar graphs.
Coloring with defect
• Mathematics
SODA '97
• 1997
This paper is concerned with algorithms and complexity results for defective coloring, where a defective (k,d)-coloring is a k coloring of the vertices of a graph such that each vertex is adjacent to
Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency
• Mathematics
J. Graph Theory
• 1986
It is proved that, for each compact surface S, there is an integer k = k(S) such that every graph in S can be (4, k)-colored; the conjecture that 4 can be replaced by 3 in this statement is conjecture.
Defective List Colorings of Planar Graphs
• Mathematics
• 1997
We combine the concepts of list colorings of graphs with the concept of defective colorings of graphs and introduce the concept of defective list colorings. We apply these concepts to vertex
On the Complexity of Some Colorful Problems Parameterized by Treewidth
• Mathematics
COCOA
• 2007
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.
A note on defective colorings of graphs in surfaces
Cowen, Cowen, and Woodall proved that, for each compact surface S, there exists an integer k = k(S) such that every graph in S can be (4, k)-colored, and conjectured that the 4 could be replaced by 3.
Efficient vertex- and edge-coloring of outerplanar graphs
• Mathematics
• 1986
The problems of finding values of the chromatic number and the chromatic index of a graph are NP-hard even for some restricted classes of graphs. Every outerplanar graph has an associated tree
Colouring Planar Graphs With Three Colours and No Large Monochromatic Components
• Mathematics
Combinatorics, Probability and Computing
• 2014
We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree Δ can be 3-coloured in such a way that each monochromatic
The Graph Coloring Problem: A Bibliographic Survey
• Mathematics
• 1998
In this chapter G = (V,E) denotes an arbitrary undirected graph without loops, where V = {v 1, v 2,…, v n } is its vertex set and E = {e 1,e 2,…, e m } ⊂ (E ×E) is its edge set. Two edges are