Partitioning and coloring graphs with degree constraints

We prove that if G is a vertex-critical graph with χ ( G ) ≥ Δ ( G ) + 1 − p ≥ 4 for some p ∈ N and ω ( H ( G ) ) ≤ χ ( G ) + 1 p + 1 − 2 , then G = K χ ( G ) or G = O 5 . Here H ( G ) is the subgraph of G induced on the vertices of degree at least χ ( G ) . This simplifies the proofs and improves the results in the paper of Kostochka, Rabern and Stiebitz (2012) [8].