We know jackshit about Mihawk's backstory. I would not bat an eye if either Nasjuro or Garling were Mihawk's father and he just took on his mother's name in a similar vein to Ace.
Mihawk, having been raised as a Holy Knight, beating his teachers and fellow students, and dipping once he discovered no other Celestial Dragon could keep up with him, going out to fight Shanks (the one Celestial Dragon kid from roughly his generation he has yet to fight) would make perfect sense to me.
The ONLY facts we know about Mihawk's past are that he was 18-19 when Roger died, that he attended his funeral, and that later on he would enter a rivalry with Shanks. Him leaving the holy knights at this time to check out Roger and running across Shanks while having a vacation on the lower realm and realizing he was a worthy rival at the time would check out very well in my mind.
Now that we know people like Mjosgard can get executed for being too out of line, even Mihawk being designated dead or alive on his bounty poster works with this theory. Now, this does not mean that this is how it went down, but is just meant to illustrate that Mihawk could very well have encountered and beaten both. Would also explain why both are still alive. Mihawk does not seem like the type of guy to kill his teachers. Despite Zoros comments about his humanity, I could easily see him sparing Garlign and Ethan in this scenario.
The more that I think about it, it could give his character some nice backstory if he was Garling's son and Shank's half-brother via some slave with the name Dracule that Garlang had. Garling's heir gets kidnapped by Pirates, and thus he starts paying attention to his bastard son, who at this point might have even been a slave too. Mihawk is roughly four or five by the time God Valley happens, so it could make sense that Garlign goes after the next best thing once Shanks is gone and starts paying attention to him.
Would give Mihawk's story and his relationship to Shanks some nice tragedy. It would also explain why Mihawk is a bit of a loner and does not easily trust others if his main relationships in life were with the bratty Holy Knight recruits he wrecked and his bum of a father who only paid attention to him once his heir was gone.