The minimum orientable genus of the repeated Cartesian product of graphs

TL;DR

This paper investigates the minimum genus of graphs formed by Cartesian products, especially focusing on hypercube-like structures combined with cycles and paths, which is key for understanding their topological complexity.

Contribution

It provides explicit calculations of the genus for Cartesian products of hypercubes with cycles and paths, advancing knowledge on graph embeddings in surfaces.

Findings

Genus of Cartesian product of 2r-cube with cycles determined

Genus of Cartesian product of 2r-cube with paths determined

Results aid in understanding non-planarity of complex network models

Abstract

Determining the minimum genus of a graph is a fundamental optimisation problem in the study of network design and implementation as it gives a measure of non-planarity of graphs. In this paper, we are concerned with determining the smallest value of $g$ such that a given graph $G$ has an embedding on the orientable surface of genus $g$. In particular, we consider the Cartesian product of graphs since this is a well studied graph operation which is often used for modelling interconnection networks. The $s$-cube $Q_i^{(s)}$ is obtained by taking the repeated Cartesian product of $i$ complete bipartite graphs $K_{s,s}$. We determine the genus of the Cartesian product of the $2r$-cube with the repeated Cartesian product of cycles and of the Cartesian product of the $2r$-cube with the repeated Cartesian product of paths.