Gonality of complete graphs with a small number of omitted edges

TL;DR

This paper determines the gonality of certain complete graphs with specific edges removed and explores their liftability to algebraic curves with matching gonality, using harmonic morphisms and plane curve models.

Contribution

It provides explicit gonality calculations for graphs formed by removing edges from complete graphs and analyzes their liftability to algebraic curves.

Findings

Gonality of graphs with edges forming a $K_h$ removed from $K_d$ is computed.

Graphs with up to $d-2$ edges removed can be lifted to curves with the same gonality using plane models.

Removal of $d-1$ edges not forming a $K_3$ requires harmonic morphisms for lifting, as plane models are insufficient.

Abstract

Let $K_d$ be the complete metric graph on $d$ vertices. We compute the gonality of graphs obtained from $K_d$ by omitting edges forming a $K_h$, or general configurations of at most $d-2$ edges. We also investigate if these graphs can be lifted to curves with the same gonality. We lift the former graphs and the ones obtained by removing up to $d-2$ edges not forming a $K_3$ using models of plane curves with certain singularities. We also study the gonality when removing $d-1$ edges not forming a $K_3$. We use harmonic morphism to lift these graphs to curves with the same gonality because in this case plane singular models can no be longer used due to a result of Coppens and Kato.