Defective coloring revisited

Cowen, Lenore; Goddard, Wayne; Jesurum, C. Esther

doi:10.1002/(sici)1097-0118(199703)24:3<205::aid-jgt2>3.0.co;2-t

Cited by 112 publications

(103 citation statements)

References 30 publications

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“…We observe that for any k, defective (k, 1)-coloring is equivalent to strongly P 3 -free k-coloring, and hence we derive the following proposition. PROPOSITION 1.2 [15].…”

Section: Previous Resultsmentioning

confidence: 92%

“…A defective (k, d)-coloring of a graph is a k-coloring in which each color class induces a subgraph with maximum degree at most d. Defective colorings have been studied for example by Archdeacon [3], by Cowen et al [14], and by Frick and Henning [19]. Cowen et al [15] have shown that the defective (3, 1)-coloring problem and the defective (2, d)-coloring problem for any d ≥ 1 are NP-hard even for planar graphs. We observe that for any k, defective (k, 1)-coloring is equivalent to strongly P 3 -free k-coloring, and hence we derive the following proposition.…”

Section: Previous Resultsmentioning

confidence: 99%

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Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult

et al. 2005

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We consider the problem of coloring a planar graph with the minimum number of colors so that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem.We present a complete picture for the case with a single forbidden connected (induced or noninduced) subgraph. The 2-coloring problem is NP-hard if the forbidden subgraph is a tree with at least two edges, and it is polynomially solvable in all other cases. The 3-coloring problem is NP-hard if the forbidden subgraph is a path with at least one edge, and it is polynomially solvable in all other cases. We also derive results for several forbidden sets of cycles. In particular, we prove that it is NP-complete to decide if a planar graph can be 2-colored so that no cycle of length at most 5 is monochromatic. Introduction.We denote by G = (V, E) a finite undirected and simple graph with |V | = n vertices and |E| = m edges. For any nonempty subset W ⊆ V , the subgraph of G induced by W is denoted by G[W ]. A clique of G is a nonempty subset C ⊆ V such that all the vertices of C are mutually adjacent. A nonempty subset I ⊆ V is independent if no two of its elements are adjacent. An r -coloring of the vertices of G is a partition V 1 , V 2 , . . . , V r of V ; the r sets V j are called the color classes of the r -coloring. An r -coloring is proper if every color class is an independent set. The chromatic number χ(G) is the minimum integer r for which a proper r -coloring of G exists.Evidently, an r -coloring is proper if and only if for every color class V j , the induced subgraph G[V j ] does not contain a subgraph isomorphic to P 2 . (We use P k to denote the path on k vertices.) This observation leads to a number of interesting generalizations of the classical graph coloring concept. One such generalization was suggested by Harary

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“…We observe that for any k, defective (k, 1)-coloring is equivalent to strongly P 3 -free k-coloring, and hence we derive the following proposition. PROPOSITION 1.2 [15].…”

Section: Previous Resultsmentioning

confidence: 92%

Section: Previous Resultsmentioning

confidence: 99%

Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult

et al. 2005

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“…Note that χ 0 (G) = χ (G). This parameter has been studied in many papers, including [Cowen et al 1986;1997;Eaton and Hull 1999]. In this paper we introduce a relaxation to vertex coloring which forbids monochromatic (k +1)-cliques, where a k-clique is a set of k pairwise-adjacent vertices.…”

Section: Introductionmentioning

confidence: 99%

Clique-relaxed graph coloring

Dunn

Nordstrom

Naymie

et al. 2011

Involve

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“…Most obviously, proper colorings are the case that F D K 2 and A D f2g. Thereafter, probably most studied is the case of coloring the vertices without creating some monochromatic subgraph, such as a star; these are often called defective colorings (see for example [1][2][3][4]). Defective colorings correspond to RASH colorings where A D f2; 3; : : : ; jF jg (where we use jF j to denote the order of F ).…”

Section: Introductionmentioning

confidence: 99%

Coloring subgraphs with restricted amounts of hues

Goddard

Melville

2017

Open Mathematics

Self Cite

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Abstract:We consider vertex colorings where the number of colors given to specified subgraphs is restricted. In particular, given some fixed graph F and some fixed set A of positive integers, we consider (not necessarily proper) colorings of the vertices of a graph G such that, for every copy of F in G, the number of colors it receives is in A. This generalizes proper colorings, defective coloring, and no-rainbow coloring, inter alia. In this paper we focus on the case that A is a singleton set. In particular, we investigate the colorings where the graph F is a star or is 1-regular.

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Defective coloring revisited

Cited by 112 publications

References 30 publications

Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult

Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult

Clique-relaxed graph coloring

Coloring subgraphs with restricted amounts of hues

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