Abstract
A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. It is known that for any two graphs G and H, \({b(G \square H) \geq {\rm {max}} \{b(G), b(H)\}}\) , where \({\square}\) stands for the Cartesian product. In this paper, we determine some families of graphs G and H for which strict inequality holds. More precisely, we show that for these graphs G and H, \({b(G \square H) \geq b(G) + b(H) - 1}\) . This generalizes one of the results due to Kouider and Mahéo.
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Balakrishnan, R., Raj, S.F. & Kavaskar, T. b-Chromatic Number of Cartesian Product of Some Families of Graphs. Graphs and Combinatorics 30, 511–520 (2014). https://doi.org/10.1007/s00373-013-1285-0
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DOI: https://doi.org/10.1007/s00373-013-1285-0