Graphs with chromatic number close to maximum degree

Let G be a color-critical graph with chi (G) >= Delta(G) = 2t + 1 >= 5 such that the subgraph of G induced by the vertices of degree 2t + 1 has clique number at most t - 1. We prove that then either t >= 3 and G = K2t+2 or t = 2 and G epsilon {K-6, O-5}, where O-5 is a special graph with chi(O-5) = 5 and vertical bar O-5 vertical bar = 9. This result for t >= 3 improves a case of a theorem by Rabern (2012)[9] and for t = 2 answers a question raised by Kierstead and Kostochka (2009) in [6]. (C) 2011 Elsevier B.V. All rights reserved.