I think Mihawk’s parents are Rayleigh and Shakki
1)Shakki ended her pirate career at the time when Mihawk was 1 year old
2)Another part of the similarity comes from one of the inspirers of Mihawk’s design:Dracula-the immortal vampire Shakky herself is 64 years old and she shows no signs of aging.
3)Mihawk’s vivre card stated that “he can see the true strength of those who stand in front of him,” and Shakky demonstrated a similar ability when compared the strength of Luffy and Rayleigh.
4)If Mihawk is Rayleigh’s son it’s obvious that he can’t use father’s surname because he is from Roger’s crew but we don’t know Shakky’s full name
5)Mihawk has a similar facial hair with Rayleigh + a little bit from Shakky and her hair color.
6)Coincidence or not,but Mihawk and prime Rayleigh have a similar patterns on their clothes.
7)Mihawk and Rayleigh were using almost the same training methods. When Mihawk was teaching Zoro how to use armament haki,he banned Zoro from drinking alcohol. Rayleigh restricted Luffy from eating,when was teaching him observation haki.
8)This can also explain why Mihawk became a swordsman, he just followed in his father’s footsteps. If its true,than this also can explain why Mihawk wanted to become the strongest swordsman(his card stated that he challenged more and more powerful foes),he decided to surpass Rayleigh,who couldn’t achive this(because of Roger,who was stronger)
9)Mihawk was introduced in chapter 50 and Rayleigh in chapter 500
10)Only people like”The dark king” and the former pirate empress could create the world strongest swordsman.
Rayleigh and Shakky together at least since Roger’s death and still don’t have kids,it’s very strange and unbelievable for me