Skin allergy= vegetarian= improvement in skin

I think there is a very huge assumption here which i am very skeptical of- the dog's allergy is due to the meat.

I agree that there may be a possiblilty but i believe that it should only be a problem for a small number of dogs. Check out these findings that involved 10 different studies and 253 dogs:

1)Beef, dairy and wheat account for 65% of all reported cases of food allergies.
2)Chicken, eggs, lamb and soy account for 25% of all reported cases of food allergies.
3) 2.4% cases were allergic to corn.

Findings from Roudebush Guilford Shanley (2000) Adverse reactions to food-Small Animal Clinical Nutrition (4th edition)

I feel going vegetarian immediately if your dog has a skin allergy may be too drastic as the allergy could be due to other causes other then meat. It should be diagnosed by the vet properly first.