Vertex-Coloring with Defects

Abstract

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 $\kappa$ colors in which the monochromatic components are acyclic and have small diameter, namely we consider (edge, $\kappa$)-colorings, in which the monochromatic components have diameter $1$, and (star, $\kappa$)-colorings, in which they have diameter $2$. We prove that the (edge, $3$)-coloring problem remains NP-complete even for graphs with maximum vertex-degree $6$, hence answering an open question posed by Cowen et al. [Cowen, Goddard, Jesurum, J. Graph Theory, 1997], and for planar graphs with maximum vertex-degree $7$, and we prove that the (star, $3$)-coloring problem is NP-complete even for planar graphs with bounded maximum vertex-degree. On the other hand, we give linear-time algorithms for testing the existence of (edge, $2$)-colorings and of (star, $2$)-colorings of partial $2$-trees. Finally, we prove that outerpaths, a notable subclass of outerplanar graphs, always admit (star, $2$)-colorings.

(Star, k)-colourings of graphs with bounded treewidth

“…In 2017, Angelini et al showed that deciding whether a planar graph with bounded maximum degree admits a (star, 3)-colouring is also NP-complete. Furthermore, they showed that there is a lineartime algorithm that decides whether there exists a (star, 2)-colouring on partial 2-trees [2].…”

On the Computational Complexity of the Bipartizing Matching Problem

“…A (k, d)-coloring of a graph G is a k-coloring of V (G) such that each vertex has at most d neighbors with same color as itself. Defective colorings were introduced independently by Andrews and Jacobson [4], Harary and Jones [36], and Cowen et al [23], which received wide attention in the literature [5,6,10,24,29]. We can see that any proper coloring is a (k, 0)-coloring, for some k ≥ 1.…”

“…Cowen et al [24] proved that it is NP-complete to determine whether a given graph is (2, 1)colorable, even for graphs of maximum degree 4 and even for planar graphs of maximum degree 5. Angelini et al [5] presented a linear-time algorithm which determines that partial 2-trees, a subclass of planar graphs, are (2, 1)-colorable. We emphasize that a k-tree has treewidth at most k, for any k ≥ 1.…”

Sobre (2,1)-colorações em grafos exoplanares maximais

“…Nesse trabalho, os autores demonstram, ainda, que esses limites são justos, ou seja: existe pelo menos um grafo planar que não possui uma (3, 1)-coloração ou uma (2, d)-coloração para todo d ≥ 0; e existe pelo menos um grafo exoplanar que não possui uma (2, 1)-coloração. Decidir a existência de (3, 1)-colorações ou (2, d)-colorações, para d > 0, são problemas NP-completos mesmo no contexto de grafos planares (Angelini et al, 2017). Ainda assim, (3, 1)-colorações foram construídas para algumas famílias de grafos planares, como os grafos planares sem triângulos adjacentes e sem ciclos de tamanho cinco (Xu, 2009), grafos planares sem triângulos adjacentes a ciclos de tamanho três ou seis (Huang, 2020), grafos planares sem ciclos de tamanho quatro ou nove (Dai et al, 2017) e grafos planares sem ciclos adjacentes de tamanho no máximo cinco (Zhang et al, 2016).…”