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The b-Continuity of Graphs with Large Girth

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Abstract

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of G is the maximum integer \(b(G)\) for which G has a b-coloring with \(b(G)\) colors. A graph G is b-continuous if G has a b-coloring with k colors, for every integer k in the interval \([\chi (G),b(G)]\). It is known that not all graphs are b-continuous, and that it is NP-complete to decide whether a given graph G is b-continuous even if \(\chi (G)\) and \(b(G)\) are known. Also, there are many results that show that finding b-colorings of graphs with large girth is an easier task. For instance, finding \(b(G)\) can be done in polynomial time when G has girth at least 7; also, regular graphs with girth at least 8 are b-continuous. In this article, we show that if G has girth at least 10, then G is b-continuous.

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Notes

  1. The graph terminology used in this paper follows [6].

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Authors and Affiliations

  1. Departamento de Computação, Universidade Federal do Ceará, Fortaleza, Brazil

    Cláudia Linhares Sales

  2. Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Brazil

    Ana Silva

Authors
  1. Cláudia Linhares Sales
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  2. Ana Silva
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Correspondence to Ana Silva.

Additional information

All authors are members of the ParGO Research Group— Parallelism, Graphs and Optimization.

This work was partially supported by CNPq and CAPES, Brazil.

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Sales, C.L., Silva, A. The b-Continuity of Graphs with Large Girth. Graphs and Combinatorics 33, 1139–1146 (2017). https://doi.org/10.1007/s00373-017-1828-x

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  • DOI: https://doi.org/10.1007/s00373-017-1828-x

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