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

The crossing number of a graph G in surface Σ, denoted by crΣ(G), is the minimum pairwise intersections edges drawing Σ. Let k be an integer satisfying k≥3, generalized Petersen P(3k,k) with vertex set V(P(3k,k))={ui,vi∣i=1,2,…,3k} and edge E(P(3k,k))={uiui+1,uivi,vivk+i∣i=1,2,…,3k}, subscripts are read modulo 3k. In this paper we investigate projective plane. We determine exact value crN1(P(3k,k)) k–2 when 3≤k≤7, moreover, for k≥8, get that k−2≤crN1(P(3k,k))≤k−1.