License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2019.34
URN: urn:nbn:de:0030-drops-109784
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10978/
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Jaffke, Lars ; Lima, Paloma T.

A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs

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Abstract

A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors. The b-chromatic number of a graph G, denoted by chi_b(G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on chi_b(G): The maximum degree Delta(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has i vertices of degree at least i-1. We obtain a dichotomy result for all fixed k in N when k is close to one of the two above mentioned upper bounds. Concretely, we show that if k in {Delta(G) + 1 - p, m(G) - p}, the problem is polynomial-time solvable whenever p in {0, 1} and, even when k = 3, it is NP-complete whenever p >= 2. We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree Delta(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by Delta(G). Second, we show that b-Coloring{} is FPT parameterized by Delta(G) + l_k(G), where l_k(G) denotes the number of vertices of degree at least k.

BibTeX - Entry

@InProceedings{jaffke_et_al:LIPIcs:2019:10978,
  author =	{Lars Jaffke and Paloma T. Lima},
  title =	{{A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{34:1--34:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10978},
  URN =		{urn:nbn:de:0030-drops-109784},
  doi =		{10.4230/LIPIcs.MFCS.2019.34},
  annote =	{Keywords: b-Coloring, b-Chromatic Number}
}

Keywords: b-Coloring, b-Chromatic Number
Collection: 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
Issue Date: 2019
Date of publication: 20.08.2019

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