The minimum orientable genus of the repeated Cartesian product of graphs
Marietta Galea, John Baptist Gauci
TL;DR
The paper determines the minimum orientable genus for the repeated Cartesian product of graphs formed from the $2r$-cubes with cycles and with paths. It uses the White-Pisanski embedding framework and Euler's formula to construct quadrilateral embeddings and derive explicit genus expressions. The main results give closed-form genus formulas: $g\left( (\Box_{\alpha=1}^{i} K_{2r,2r}) \Box (\Box_{\alpha=1}^{j} C_{2m_{\alpha}})\right) = 1 + M^{(j)} 2^{2i+j-2} r^{i} (j + i r - 2)$ and $g\left( (\Box_{\alpha=1}^{i} K_{2r,2r}) \Box (\Box_{\alpha=1}^{j} P_{2m_{\alpha}})\right) = 1 + 2^{2i+j-3} r^{i} M^{(j)} (2 i r + 2 j - m^{(j)} - 4)$, with related corollaries. These findings extend prior work by White and Pisanski, broadening exact genus computations for structured product graphs relevant to network design.
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.