An upper bound for the crossing number of augmented cubes

Abstract

A good drawing of a graph G is a drawing where the edges are non-self-intersecting and each two edges have at most one point in common, which is either a common end vertex or a crossing. The crossing number of a graph G is the minimum number of pairwise intersections of edges in a good drawing of G in the plane. The n-dimensional augmented cube AQn, proposed by S.A. Choudum and V. Sunitha, is an important interconnection network with good topological properties and applications. In this paper, we obtain an upper bound on the crossing number of AQn less than 26 32 4 n − (2n 2 + 7 2 n − 6)2 n−2 .

A survey of graphs with known or bounded crossing numbers

“…An example of the augmented cube AQ 3 is displayed in Figure 20. In 2013, Wang et al [195] investigated the crossing number of the augmented cube and discovered lower and upper bounds: Theorem 6.8 (Wang et al, 2013 [195]) Consider the augmented cube AQ n . Then, cr(AQ 3 ) = 4, cr(AQ 4 ) ≤ 46, cr(AQ 5 ) ≤ 328, cr(AQ 6 ) ≤ 1848, cr(AQ 7 ) ≤ 9112, and for n ≥ 8,…”