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The b-chromatic number (G) of a graph G is defined as the largest number k for which the vertices of G can be colored with k colors satisfying the following property: for each i, 1 i k, there exists a vertex x i of color i such that for all j = i, 1 j k there exists a vertex y j of color j adjacent tois the chromatic number of H. We characterize all b-perfect bipartite graphs and all b-perfect P 4 -sparse graphs by minimal forbidden induced subgraphs. We also prove that every 2K 2 -free and P 5 -free graph is b-perfect.

a b s t r a c tA b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list F of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.

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a b s t r a c tIn an article Cheng (2009) [3] published recently in this journal, it was shown that when k ≥ 3, the problem of deciding whether the distinguishing chromatic number of a graph is at most k is NP-hard. We consider the problem when k = 2. In regards to the issue of solvability in polynomial time, we show that the problem is at least as hard as graph automorphism, but no harder than graph isomorphism.

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