Recursive Structure and Bandwidth of Hales-Numbered Hypercube

TL;DR

This paper reveals recursive structures in Hales-numbered hypercubes that simplify the proof of their bandwidth and provides a closed-form solution for the hypercube antibandwidth problem, enhancing understanding of hypercube graph properties.

Contribution

The paper introduces a new recursive perspective on hypercube structures, simplifying bandwidth proofs and solving the antibandwidth problem with a closed-form expression.

Findings

Simplified proof of hypercube bandwidth formula

Closed-form solution for hypercube antibandwidth

Identification of recursive structures in hypercube adjacency matrices

Abstract

The Hales numbered $n$-dimensional hypercube and the corresponding adjacency matrix exhibit interesting recursive structures in $n$. These structures lead to a very simple proof of the well-known bandwidth formula for hypercube, whose proof was thought to be surprisingly difficult. A related problem called hypercube antibandwidth, for which Harper proposed an algorithm, is also reexamined in the light of the above recursive structures, and a close form solution is found.