The crossing number of cubes with small order

TL;DR

This paper determines the exact crossing numbers for specific small-order hypercube variants, including crossed, locally twisted, and Möbius cubes, providing precise graph drawing complexity measures.

Contribution

It provides the first exact crossing number values for these hypercube variants with small order, expanding understanding of their geometric properties.

Findings

Exact crossing numbers for crossed cube, locally twisted cube, and Möbius cube.

Improved understanding of graph drawing complexity for small hypercube variants.

Contributes to graph theory by clarifying crossing number values for specific hypercube modifications.

Abstract

The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. In this paper, we give the exact values of crossing numbers for some variations of hypercube with order at most four, including crossed cube, locally twisted cube and M\"{o}bius cube.