On the b-chromatic number of regular graphs without 4-cycle

The b-chromatic number of a graph G, denoted by @f(G), is the largest integer k for which G admits a proper coloring by k colors, such that each color class has a vertex that is adjacent to at least one vertex in each of the other color classes. We prove that, for each d-regular graph G which contains no 4-cycle, @f(G)>[email protected][email protected]?; and besides, if G has a triangle, then @f(G)>[email protected][email protected]?. Also, if G is a d-regular graph that contains no 4-cycle and diam(G)>=6, then @f(G)=d+1. Finally, we show that, for any d-regular graph G which does not contain 4-cycle and has vertex connectivity less than or equal to d+12, @f(G)=d+1. Moreover, when the vertex connectivity is less than 3d-34, we introduce a lower bound for the b-chromatic number in terms of the vertex connectivity.