Multiplicity-Free Gonality on Graphs

TL;DR

This paper introduces the concept of multiplicity-free gonality in graphs, compares it with divisorial gonality, and explores its computational complexity and properties across different graph families.

Contribution

It defines multiplicity-free gonality, establishes conditions for equality with divisorial gonality, and proves its NP-hardness to compute.

Findings

Multiplicty-free gonality can differ significantly from divisorial gonality.

No function of gonality bounds multiplicity-free gonality.

NP-hardness of computing multiplicity-free gonality.

Abstract

The divisorial gonality of a graph is the minimum degree of a positive rank divisor on that graph. We introduce the multiplicity-free gonality of a graph, which restricts our consideration to divisors that place at most \(1\) chip on each vertex. We give a sufficient condition in terms of vertex-connectivity for these two versions of gonality to be equal; and we show that no function of gonality can bound multiplicity-free gonality, even for simple graphs. We also prove that multiplicity-free gonality is NP-hard to compute, while still determining it for graph families for which gonality is currently unknown. We also present new gonalities, such as for the wheel graphs.