An upper bound for the crossing number of augmented cubes

TL;DR

This paper establishes an upper bound on the crossing number of the n-dimensional augmented cube, a key interconnection network, providing insights into its topological complexity and drawing properties.

Contribution

The paper derives a new upper bound for the crossing number of augmented cubes, enhancing understanding of their geometric and topological characteristics.

Findings

Upper bound less than 26/324^n minus a quadratic term times 2^{n-2}

Provides theoretical limits on the crossing number for augmented cubes

Advances knowledge of interconnection network topology

Abstract

A {\it 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 {\it 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 {\it $n$-dimensional augmented cube} $AQ_n$, 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 $AQ_n$ less than $26/324^{n}-(2n^2+7/2n-6)2^{n-2}$.