On the gonality, treewidth, and orientable genus of a graph

TL;DR

This paper explores relationships between graph gonality, treewidth, and genus, revealing that hyperelliptic graphs are planar and introducing bielliptic graphs that embed into genus-one surfaces, along with constructions for graphs of higher gonality.

Contribution

It establishes new connections between gonality, treewidth, and genus, including classifications of hyperelliptic and bielliptic graphs and constructions for graphs of higher gonality.

Findings

Hyperelliptic graphs are planar.

Bielliptic graphs embed into genus-one surfaces.

Existence of trigonal graphs with specified properties for all g ≥ 0.

Abstract

We examine connections between the gonality, treewidth, and orientable genus of a graph. Especially, we find that hyperelliptic graphs in the sense of Baker and Norine are planar. We give a notion of a bielliptic graph and show that each of these must embed into a closed orientable surface of genus one. We also find, for all $g\ge 0$, trigonal graphs of treewidth 3 and orientable genus $g$, and give analogues for graphs of higher gonality.