Graphs with chromatic number close to maximum degree

Let G be a color-critical graph with χ ( G ) ≥ Δ ( G ) = 2 t + 1 ≥ 5 such that the subgraph of G induced by the vertices of degree 2 t + 1 has clique number at most t − 1 . We prove that then either t ≥ 3 and G = K 2 t + 2 or t = 2 and G ∈ { K 6 , O 5 } , where O 5 is a special graph with χ ( O 5 ) = 5 and | O 5 | = 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].