On graphs G for which both G and G are claw-free

Let G be a graph with |V (G)| ≥ 10. We prove that if both G and G are claw-free, then min{∆(G), ∆(G)} ≤ 2. As a generalization of this result in the case where |V (G)| is sufficiently large, we also prove that if both G and G are K1,t-free, then min{∆(G), ∆(G)} ≤ r(t− 1, t)− 1 where r(t− 1, t) is the Ramsey number.