Act 1: Mihawk meets Shanks and rejects a direct duel.
Act 2: Mihawk goes to Marineford war, states he doesn't know the difference between his power and a top tier, he is stopped by a mere subordinate (Vista) of that top tier. Suddenly Shanks appears and he runs away before the war was even stopped by Sengoku.
Act 3: Mihawk had to change his island because marines were after him and allied with Crocodile.
Act 4: Mihawk rejects Enperor title since he wants live in peace.
Act 5: Mihawk tells Buggy he wont fight any Emperor just because he says so.
So how is the show called? Yeah, Mihawk worlds strongest swordsman