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arXiv · 2405.05368

The minimum orientable genus of the repeated Cartesian product of graphs

Marietta Galea·John Baptist Gauci
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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.

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Marietta Galea, John Baptist Gauci. 2024-05-08. The minimum orientable genus of the repeated Cartesian product of graphs. https://arxiv.org/abs/2405.05368

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