Subcolorings and the subchromatic number of a graph

Gimbel, John; Hartman, Chris

doi:10.1016/s0012-365x(03)00177-8

Cited by 24 publications

(17 citation statements)

References 12 publications

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“…It has been studied by Broere and Mynhardt [4], by Albertson et al [2], by Fiala et al [18], or Gimbel and Hartman [21], and by Broersma et al [6]. We further utilize especially the following result: PROPOSITION 1.1 [18].…”

Section: Previous Resultsmentioning

confidence: 95%

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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Section: Previous Resultsmentioning

confidence: 95%

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

et al. 2005

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“…Many variants and generalizations of basic concepts have been introduced and intensively studied over the years. Since this subject is too wide to be surveyed in a short paper, we mention just a few examples like subcoloring known also as P 3 -free coloring (where P p denotes the chordless path on p vertices), P 4 -free coloring and improper coloring, and we refer to appropriate literature on other variants, e.g., many results on subcoloring can be found in Albertson et al [2], Broere and Mynhardt [8], Fiala et al [16] as well as in work of Gimbel and Hartman [17]. For results on P 4 -free coloring see, e.g., Gimbel and Nešetřil [18] and a paper of Hoàng and Le [23], while for improper coloring we refer the reader to papers of Bermond et al [4], Cowen et al [15] and Havet et al [21].…”

Section: Related Research and Our Resultsmentioning

confidence: 99%

Dynamic F-free Coloring of Graphs

Borowiecki

Sidorowicz

2018

Graphs and Combinatorics

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A problem of graph F-free coloring consists in partitioning the vertex set of a graph such that none of the resulting sets induces a graph containing a fixed graph F as an induced subgraph. In this paper we consider dynamic F-free coloring in which, similarly as in online coloring, the graph to be colored is not known in advance; it is gradually revealed to the coloring algorithm that has to color each vertex upon request as well as handle any vertex recoloring requests. Our main concern is the greedy approach and characterization of graph classes for which it is possible to decide in polynomial time whether for the fixed forbidden graph F and positive integer k the greedy algorithm ever uses more than k colors in dynamic F-free coloring. For various classes of graphs we give such characterizations in terms of a finite number of minimal forbidden graphs thus solving the above-mentioned problem for the so-called F-trees when F is 2-connected, and for classical trees, when F is a path of order 3 (the latter variant is also known as subcoloring or 1-improper coloring).

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“…Note that in a triangle-free graph, a vertex 2-subcoloring corresponds to a vertex partition into two forests with maximum degree one. The problem Vertex 2-Subcolorability is NP-complete even for triangle-free planar graphs with maximum degree 4 [11,14].…”

Section: Spider Arboricity Of Bipartite Graphsmentioning

confidence: 99%