The crossing numbers of generalized Petersen graphs with small order

The generalized Petersen graph P(n,k)P(n,k) is an undirected graph on 2n2n vertices with V(P(n,k))={ai,bi:0≤i≤n−1}V(P(n,k))={ai,bi:0≤i≤n−1} and E(P(n,k))={aibi,aiai+1,bibi+k:0≤i≤n−1,subscripts modulo n}E(P(n,k))={aibi,aiai+1,bibi+k:0≤i≤n−1,subscripts modulo n}. Fiorini claimed to have determined the crossing numbers of P(n,3)P(n,3) and showed all the values of cr(P(n,k))cr(P(n,k)) for nn up to 14, except 12 unknown values. Lovrečič Saražin proved cr(P(10,4))=cr(P(10,6))=4cr(P(10,4))=cr(P(10,6))=4. Richter and Salazar found a gap in Fiorini’s paper, which invalidated his principal results about cr(P(n,3))cr(P(n,3)), and gave the correct proof for cr(P(n,3))cr(P(n,3)). In this paper, we show the crossing numbers of all P(n,k)P(n,k) for nn up to 16.