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  • 1 Pondicherry University, Pondicherry–605014, India
  • 2 CIT campus, Taramani, Chennai-600113, India
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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 was conjectured in [10], that for any two graphs G and H, b(G[H]) ≦ b(G) − 1|V (H)| + Δ(H) + 1 and b(GH) ≦ max {b(G)(Δ(H) + 1), b(H) Δ(G) + 1)}, where G[H] and GH denotes the lexicographic product and the strong product of G and H, respectively. In this paper, we disprove both conjectures.

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