No. The American Pit Bull Terrier is a breed and the "Gator pit" or "Gator mouth pit" is a term used by backyard breeders to sell puppies. There are two bloodlines of APBT that could be considered "gator". Those are CH Crenshaw's Gator ROM and CH Plumber's Alligator POR. Don't buy from anyone who's saying their dog is a Texas Red Nose, Gator pit, Gator mouth pit, "rare" blue, or a lock jaw pit. To be safe, anyone looking for pups should shy away from someone breeding very young dogs, having many litters and anyone who can't get basic things right. It is an American Pit Bull Terrier. Not a pitt bull, American Pitt Bull, Pitt Bull Terrier. If someone ever says they have "rare" blue dogs, they are full of it. That is the most over bred color there is. Rednose/blue nose/black nose/ doesn't mean anything. That is decided by the level of pigment in the dog's skin. It has nothing to do with the quality of the dog. And there is no tiger striped pitt bull, it's just a basic brindled dog.
Here are the pedigrees on those two legit "gator" dogs mentioned earlier.
CH Crenshaw's Gator ROM - http://www.apbt.online-pedigrees.com/public/printPedigree.php?dog_id=110
CH Plumber's Alligator POR - http://www.apbt.online-pedigrees.com/public/printPedigree.php?dog_id=380