Abstract
A b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The b-chromatic number of a graph G, denoted by χ b (G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is b-continuous if it admits a b-coloring with t colors, for every \(t = \chi(G), \ldots, \chi_b(G)\) . We define a graph G to be b-monotonic if χ b (H 1) ≥ χ b (H 2) for every induced subgraph H 1 of G, and every induced subgraph H 2 of H 1. In this work, we prove that P 4-sparse graphs (and, in particular, cographs) are b-continuous and b-monotonic. Besides, we describe a dynamic programming algorithm to compute the b-chromatic number in polynomial time within these graph classes.
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Flavia Bonomo: Partially supported by ANPCyT PICT-2007-00533 and PICT-2007-00518, and UBACyT Grants X069 and X606 (Argentina).
Guillermo Durán: Partially supported by FONDECyT Grant 1080286 and Millennium Science Institute “Complex Engineering Systems” (Chile), and ANPCyT PICT-2007-00518 and UBACyT Grant X069 (Argentina).
Javier Marenco: Partially supported by ANPCyT PICT-2007-00518 and UBACyT Grant X069 (Argentina).
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Bonomo, F., Durán, G., Maffray, F. et al. On the b-Coloring of Cographs and P 4-Sparse Graphs. Graphs and Combinatorics 25, 153–167 (2009). https://doi.org/10.1007/s00373-008-0829-1
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DOI: https://doi.org/10.1007/s00373-008-0829-1