Chromatic graph with disjoints subgraphs

How to prove that a graph with chromatic number $$geq 4$$ has in it two disjoint by vertices subgraphs with a chromatic number of $$2$$ and $$3$$.

Also there is no $$K_4$$ in the original graph

I was thinking about proving by construction, but it doesn’t seem to be working.

Mathematics Asked on January 3, 2021

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