A note on circular wirelength for hypercubes

TL;DR

This paper proves that the minimal circular wirelength embedding of an n-dimensional hypercube into a circuit is achieved by Gray coding, confirming the CT conjecture and addressing gaps in previous proof attempts.

Contribution

It provides a complete proof that Gray coding minimizes circular wirelength for hypercube embeddings, resolving prior uncertainties.

Findings

Gray coding attains minimum circular wirelength for hypercubes

The CT conjecture is confirmed with a rigorous proof

Gaps in Guu's previous proof are eliminated

Abstract

We study embeddings of the $n$-dimensional hypercube into the circuit with $2^n$ vertices. We prove that the circular wirelength attains minimum by gray coding, which is called the CT conjecture by Chavez and Trapp (Discrete Applied Mathematics, 1998). This problem had claimed to be settled by Ching-Jung Guu in her doctor dissertation "The circular wirelength problem for hypercubes" (University of California, Riverside, 1997). Many people argue there are gaps in her proof. We eliminate gaps in her dissertation.