Embedding Grid Graphs on Surfaces

TL;DR

This paper studies how grid graphs can be embedded on surfaces, providing formulas and bounds for their genus, and classifies which grid graphs are planar or toroidal based on their combinatorial properties.

Contribution

It introduces new methods to determine the genus of grid graphs and offers a complete classification of planar and toroidal grid graphs.

Findings

Determined the genus of large classes of k-dimensional grid graphs.

Established bounds for the genus of 3D grid graphs.

Classified all planar and toroidal grid graphs.

Abstract

In this paper, we analyze embeddings of grid graphs on orientable surfaces. We determine the genus of a large class of k-dimensional grid graphs and effective two-sided bounds for the genus of any 3-dimensional grid graph, both in terms of a grid graph's combinatorics. As an application, we provide a complete classification of planar and toroidal grid graphs. Our work requires a variety of combinatorial arguments to determine effective lower bounds on the genus of a grid graph, along with explicitly constructing embeddings of grid graphs on surfaces to determine effective upper bounds on their genera.