# Embedding hypercubes into torus and Cartesian product of paths and cycles for minimizing wirelength

## TL;DR

This paper proves the minimum wirelength for embedding hypercubes into Cartesian products of paths and cycles, confirming Gray code embedding as an optimal strategy for such problems.

## Contribution

It mathematically proves the conjecture on minimum wirelength and establishes Gray code embedding as optimal for hypercube embeddings into these graph products.

## Key findings

- Proved the minimum wirelength for hypercube embedding into Cartesian products of paths and cycles.
- Confirmed Gray code embedding as the optimal strategy for these embeddings.
- Solved an open problem in the embedding of hypercubes into torus-like graphs.

## Abstract

Though embedding problems have been considered for several regular graphs, it is still an open problem for hypercube into torus. In the paper, we prove the conjecture mathematically and obtain the minimum wirelength of embedding for hypercube into Cartesian product of paths and/or cycles. In addition, we explain that Gray code embedding is an optimal strategy in such embedding problems.