Graphs with chromatic number close to maximum degree

Let GG be a color-critical graph with χ(G)≥Δ(G)=2t+1≥5χ(G)≥Δ(G)=2t+1≥5 such that the subgraph of GG induced by the vertices of degree 2t+12t+1 has clique number at most t−1t−1. We prove that then either t≥3t≥3 and G=K2t+2G=K2t+2 or t=2t=2 and G∈{K6,O5}G∈{K6,O5}, where O5O5 is a special graph with χ(O5)=5χ(O5)=5 and |O5|=9|O5|=9. This result for t≥3t≥3 improves a case of a theorem by Rabern (2012) [9] and for t=2t=2 answers a question raised by Kierstead and Kostochka (2009) in [6].