On minimum vertex cover of generalized Petersen graphs

Determining the size of minimum vertex cover of a graph G, denoted by β(G), is an NP-complete problem. Also, for only few families of graphs, β(G) is known. We study the size of minimum vertex cover in generalized Petersen graphs. For each n and k (n > 2k), a generalized Petersen graph P (n, k), is defined by vertex set {ui, vi} and edge set {uiui+1, uivi, vivi+k}; where i = 1, 2, . . . , n and subscripts are reduced modulo n. First, we characterize minimum vertex covers in generalized Petersen graphs by defining a new quantity. Second, we prove that β(P (n, k)) ≥ n + (n,k)+1 2 , for all odd n, where (n, k) is the greatest common devisor of n and k; β(P (n, k)) ≤ n + k+1 2 for all odd k; for all m < k, β(P (n, k)) ≤ n mβ(P (m, k (mod m))) when m | n and otherwise β(P (n, k)) ≤ n m β(P (m, k (mod m))) + 2k. Most of these bounds are sharp. Third, we determine the exact values of β(P (n, k)) for k = 1, 3 and in the case when n is even and k is odd, also the case when both n and k are odd and k | n. We conjecture that β(P (n, k)) ≤ n + n5 , for all n and k.