Sparse graphs of high gonality

TL;DR

This paper demonstrates that graphs with low treewidth can have arbitrarily high gonality and that gonality can decrease when passing to subgraphs, addressing open problems in graph theory related to algebraic analogues.

Contribution

It proves the existence of connected graphs with treewidth 2 of arbitrarily high gonality and shows gonality can decrease in subgraphs, resolving three open problems.

Findings

Existence of connected graphs with treewidth 2 and arbitrarily high gonality

Existence of graph pairs where a subgraph has lower gonality than the original

Resolution of three open problems in graph gonality theory

Abstract

By considering graphs as discrete analogues of Riemann surfaces, Baker and Norine (Adv. Math. 2007) developed a concept of linear systems of divisors for graphs. Building on this idea, a concept of gonality for graphs has been defined and has generated much recent interest. We show that there are connected graphs of treewidth 2 of arbitrarily high gonality. We also show that there exist pairs of connected graphs $\{G,H\}$ such that $H\subseteq G$ and $H$ has strictly lower gonality than $G$. These results resolve three open problems posed in a recent survey by Norine (Surveys in Combinatorics 2015).