The crossing numbers of generalized Petersen graphs with small order

Abstract

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