Graphs with chromatic number close to maximum degree

AbstractLet G be a color-critical graph with χ(G)≥Δ(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∈{K6,O5}, where O5 is a special graph with χ(O5)=5 and |O5|=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]