On Eggleton and Guy's conjectured upper bound for the crossing number of the $n$-cube
Faria, Luerbio, de Figueiredo, Celina Miraglia Herrera · Czech digital mathematics library · 2000
The crossing number v(G) of a graph G is the smallest integer such that there is a drawing for G with v(G) crossings of edges.Let Q n denote the n-dimensional cube.Eggleton and Guy conjectured in 1970 that v(Q n ) 7, with number of crossings j §^4 n -2n2 ~12 ln+34 2 n ~2, establishing a new upper bound for v(Q n ).Our family of drawings confirms Eggleton and Guy's conjectured upper bound when n = 7 and 8.In addition, our upper bound improves the upper bound v(Q n ) < 4 n | -2 n ~3n 2 -2 n ~43 + (~2) n ^ due to Madej.