Graphs with large maximum degree containing no edge-critical graphs

We say that a graph is edge-critical if it contains an edge whose deletion reduces the chromatic number of it. Let F be an edge-critical graph with chromatic number r+2. We determine the maximum number of edges of a graph of order n with maximum degreeΔthat contains no copy of F whenΔ<= n-ß_Fn^S_F, where ß_Fand S_F<1 are constants depending on F. Furthermore, the unique extremal graph is a complete (r+1)-partite graph. Our result is a generalization of a result of Balister, Bollobas, Riordan and Schelp.