Systolic embedding of graphs on translation surfaces

TL;DR

This paper studies special graph embeddings called systolic embeddings on translation surfaces, proving their existence for all finite graphs, estimating surface genera, and exploring embeddings of specific graph types.

Contribution

It introduces the concept of systolic embeddings on translation surfaces, proves their existence for all finite graphs, and analyzes genus bounds for various graph classes.

Findings

Any finite graph admits an essential-systolic embedding on a translation surface.

The paper estimates the genera of surfaces supporting such embeddings.

It characterizes graphs that admit cellular-systolic embeddings, including wedge graphs and others.

Abstract

An embedding of a graph on a translation surface is said to be \emph{systolic} if each vertex of the graph corresponds to a singular point (or marked point) and each edge corresponds to a shortest saddle connection on the translation surface. The embedding is said to be \emph{cellular} (respectively \emph{essential}) if each complementary region is a topological disk (respectively not a topological disk). In this article, we prove that any finite graph admits an essential-systolic embedding on a translation surface and estimate the genera of such surfaces. For a wedge $\Sigma_n$ of $n$ circles, $n\geq2$, we investigate that $\Sigma_n$ admits cellular-systolic embedding on a translation surface and compute the minimum and maximum genera of such surfaces. Finally, we have identified another rich collection of graphs with more than one vertex that also admit cellular-sytolic embedding on translation surfaces.