Regular Graphs with Small Skewness and Crossing Numbers

Let G be a graph. The crossing number of G , denoted c r ( G ) , is the minimum number of pairwise intersections of its edges when G is drawn in the plane. The skewness of G , denoted s k ( G ) , is defined to be the minimum number of edges in G whose removal results in a planar graph. We show, by constructions, the existence of 5 -regular graphs on n vertices with skewness s and crossing number c for (i) s = 1 , c = 3 and n ≡ 2 ( mod 4 ) ≥ 10 , (ii) s = 2 , c = 2 and n ≥ 8 is even, and (iii) s = 1 , c = 1 and n ≥ 14 is even.