Abstract
A b-coloring of a graph is a proper coloring of its vertices such that every color class contains a vertex that has neighbors in all other color classes. The b-chromatic number of a graph is the largest integer b(G) such that the graph has a b-coloring with b(G) colors. This metric is upper bounded by the largest integer m (G) for which G has at least m (G) vertices with degree at least m (G) —1. There are a number of results reporting that graphs with high girth have high b-chromatic number when compared to m(G). Here, we prove that every graph with girth at least 8 has b-chromatic number at least m(G) 3 - 1. This proof is constructive and yields a polynomial time algorithm to find the b-chromatic number of G. Furthermore, we improve known partial results related to reducing the girth requirement of our proof.
Partially supported by FUNCAP, CAPES and CNPq — Brasil
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Campos, V., Lima, C., Silva, A. (2013). B-Coloring Graphs with Girth at Least 8. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_52
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DOI: https://doi.org/10.1007/978-88-7642-475-5_52
Publisher Name: Edizioni della Normale, Pisa
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