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

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