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The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. We prove that every d-regular graph with at least 2d 3 vertices has b-chromatic number d + 1, that the b-chromatic number of an arbitrary d-regular graph with girth g = 5 is at least d+1 2 and that every d-regular graph, d ≥ 6, with diameter at least d and with no 4-cycles admits a b-coloring with d + 1 colors.

The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring in which every color class contains at least one vertex adjacent to some vertex in all the other color classes. It is proved that with four exceptions, the b-chromatic number of cubic graphs is 4. The exceptions are the Petersen graph, K 3,3 , the prism over K 3 , and one more sporadic example on 10 vertices.

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