The crossing number of the generalized Petersen graph P(3k,k) in the projective plane

The crossing number of a graph G in a surface Sigma, denoted by cr(Sigma)(G), is the minimum number of pairwise intersections of edges in a drawing of G in Sigma. Let k be an integer satisfying k > 3, the generalized Petersen graph P(3k, k) is the graph with vertex set V(P(3k, k)) = {u(i), v(i) | i = 1, 2, . .., 3k} and edge set E(P(3k, k)) = {u(i)u(i)+1, u(i)v(i), v(i)v(k+i) | i = 1, 2, . .., 3k}, the subscripts are read modulo 3k. In this paper we investigate the crossing number of P(3k, k) in the projective plane. We determine the exact value of crN1 (P(3k, k)) is k - 2 when 3 < k < 7, moreover, for k > 8, we get that k - 2 < cr(N1) (P(3k, k)) < k -1.