Partitioning and coloring graphs with degree constraints

We prove that if GG is a vertex-critical graph with χ(G)≥Δ(G)+1−p≥4χ(G)≥Δ(G)+1−p≥4 for some p∈Np∈N and ω(H(G))≤χ(G)+1p+1−2, then G=Kχ(G)G=Kχ(G) or G=O5G=O5. Here H(G)H(G) is the subgraph of GG induced on the vertices of degree at least χ(G)χ(G). This simplifies the proofs and improves the results in the paper of Kostochka, Rabern and Stiebitz (2012) [8].