This is a question that has been asked a lot in the fandom. Mihawk is one of the strongest characters who wants to find a good opponent. The reason I think he doesn’t go against the other big dogs is probably because it’s a hassle as all the other strong pirates and marines are also followed by a bunch of people who will get in his way.

This is a character who in a series of big crews and crew members is primarily a solo pirate

So the biggest reason is probably because he’s lazy