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  • 1 University of Maribor Faculty of Natural Sciences and Mathematics Koroška cesta 160 2000 Maribor Slovenia
  • 2 University of Maribor Faculty of Electrical Engineering and Computer Science Smetanova ulica 17 2000 Maribor Slovenia
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A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer φ(G) for which a graph has a b-coloring with φ(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product GH, the lexicographic product G[H] and the direct product G × H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the b-chromatic number of PnH, CnH, Pn[H], Cn[H], and Km,n[H] can be determined for an arbitrary graph H, when integers m and n are large enough.

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