It is known that if G is P4-free, then G is (χ,Γ)-perfect [ChSe79], which implies that χ(G)=χc(G)=Γc(G)=Γ(G); so in our proof we can suppose that G has some P4. To prove Lemma 4, we pick some dominating P4 in G and an optimal coloring of G and chang…