Is Dracule Mihawk really the strongest swordsman? When Shanks entered the Battle of Marineford, it was to end it. All he used was a single sword to stop Akainu (considered as the strongest in the marines now) from killing Coby.



Mihawk left the battlefield saying that he had only agreed to fight Whitebeard. So is this just a normal reason for him to leave or was he scared that he would have lost to Shanks? Because, in the war Mihawk fought with other pirates except for Whitebeard so it seems a bit contradictory to what he said while leaving.



Then there was a scene where he actually tracks down Shanks just to inform him about Luffy's first bounty. Is there any particular reason behind this?



Also, does Mihawk has some relation with Shanks or is he scared of him?
If you look at the wiki page of Mihawk


  Dracule Mihawk is a member of the Shichibukai and the first one to
  be revealed. He is also the current holder of the title, "Greatest
  Swordsman in the World.


This should answer the title question.

As for the relationship with Shanks (it is listed under the Friends list),


  In the past, Mihawk was a rival to one of the most powerful pirates in
  One Piece, Shanks. However, Mihawk lost interest in him after Shanks
  lost his left arm. He retains a neutral, though somewhat friendly
  relationship with Shanks, and is able to track him down if the need
  arises. He and Shanks do manage to get along, regardless of the fact
  that the two are opposites in personalities. Shanks is also able to
  get Mihawk to join in when he threw a party to celebrate Luffy's
  coming, even when moments beforehand Mihawk had scorned how laid-back
  Shanks was.
  
  Mihawk even mused out an apology to Shanks during the battle at
  Marineford, where he declares that he will not be holding anything
  back against Luffy right before going after him. When Shanks
  showed up during the Whitebeard War, Mihawk left the battle, stating
  that battling Shanks was beyond the scope of the World Government's
  agreement.


So its not that he's really scared of Shanks, but just that he relishes the Frenemy kind-of relationship with him, and that's why moves out of the battle, when Shanks enters the scene.