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

Abstract

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