It's still possible he have it but even if he doesn't, it's simply explained by him not being a Leader
Haoshoku Haki is the Sign of a King & Leader
Mihawk is King of Swordsmen in terms of Power but he doesn't rule over anyone nor does he want to have any follower
(Even Perona, the way he talks to her clearly shows that this guy doesn't care about Leadership/Rulership)
Rayleigh & Katakuri & Zoro (If he shows CoC) even if they have someone Superior than them in Leadership still showed qualities of being someone worth following & also showed desire to (Rayleigh replaced Roger as Captain, Katakuri is a Role Model to his Siblings & they listen to him, Zoro wanted to lead Samurai & is seen as SH's Vice Captain)
CoC Users are those with the ability to Influence Others & Conquer their will, Mihawk only wants to prove he is Stronger/Better
Haoshoku Haki is the Sign of a King & Leader
Mihawk is King of Swordsmen in terms of Power but he doesn't rule over anyone nor does he want to have any follower
(Even Perona, the way he talks to her clearly shows that this guy doesn't care about Leadership/Rulership)
Rayleigh & Katakuri & Zoro (If he shows CoC) even if they have someone Superior than them in Leadership still showed qualities of being someone worth following & also showed desire to (Rayleigh replaced Roger as Captain, Katakuri is a Role Model to his Siblings & they listen to him, Zoro wanted to lead Samurai & is seen as SH's Vice Captain)
CoC Users are those with the ability to Influence Others & Conquer their will, Mihawk only wants to prove he is Stronger/Better



Yes, his duels are a legend, duels with characters weaker than KAIDO.