I have a theory regarding Mihawk being Rayleigh’s disciple. Both being very good swordsmen (one of them even the best in the world) is kind of a hint for me. And it would explain two still unknown things:
1. How the friendship between Shanks and Mihawk developed (Shanks, too, being a swordsman) maybe they were both his disciples at the same time.
2. Zoro’s scar on his eye. We know that Zoro’s and Rayleigh’s scar might have the same reason, which would be the training of Observation Haki. And after the War of the Best at Marineford, Zoro became Mihawk’s disciple. Mihawk knew of that technique to train Observation Haki because he saw it from Rayleigh. Mihawk having the hawkeyes did not have to train his Observation Haki but he showed that technique to Zoro.
*Theory by Oz