On the minimum vertex cover of generalized Petersen graphs

It is known that any vertex cover of the generalized Petersen graph P(n,k) has size at least n. Behsaz, Hatami and Mahmoodian characterized such graphs with minimum vertex cover numbers n and n+1, and those with k≤3. For k≥4 and n≥2k+2, we prove that if the 2-adic valuation of n is less than or equal to that of k, then the minimum vertex cover number of P(n,k) equals n+2 if and only if n∈{2k+2,3k−1,3k+1}.