Usually this fight starts a shitstorm of epic proportions arguing over titles of worlds strongest swordsman vs yonkou, BUT, since you posted a picture of young Mihawk, Shanks wtfstomps :D
Well this depends on a lot, mihawk alone couldn't defeat whitebeard shanks casually clashed swords with him and it split the sky. Mihawk was at the war and nobody cared and continued to fight him regardless, Shanks came and everyone stopped and began to cower in fear even though they had mihawk on their side.
Their was no proof of shanks beating Mihawk, other than they has countless battles with each other when shanks had his arm.. Since he lost it Mihawk lost intrest in fighting him since he thinks shawnks would be at a disavantage if they fought again(but i think shanks could still take him)
As well as this Mihawk's still holds the title of the 'Worlds Greatest Swordsman' and last I checked Shanks uses a sword.
Mihawk also was one of the very few people who escaped the entire Marineford War untouched and he fought a couple of upper tier characters.
Shanks doesn't have to want to be the world's greatest swordsman just because he wields a sword. I'm 100% sure if rayleigh wanted mihawk would have to hand over his sword and eyes to give birth to hawk eyes rayleigh. Not all swordsman have to fight mihawk especially those who don't want such a title.
Years after "leaving" the show he helped create, Darabont wants his cut.