A note on Brill--Noether existence for graphs of low genus

TL;DR

This paper investigates Baker's Brill--Noether existence conjecture for graphs of low genus, providing new combinatorial proofs and confirming the conjecture for specific cases and graph families.

Contribution

It offers the first combinatorial proof of the Brill--Noether existence theorem for metric graphs in certain genera and verifies the conjecture for graphs of genus up to 5.

Findings

All graphs of genus ≤ 5 admit divisors of specified rank and degree when the Brill--Noether number is non-negative.

The conjecture holds in rank 1 for certain highly connected graph families of arbitrary genus.

First combinatorial proof of Brill--Noether existence theorem for metric graphs in relevant genera.

Abstract

In an influential 2008 paper, Baker proposed a number of conjectures relating the divisor theory of algebraic curves with an analogous combinatorial theory on finite graphs. In this note, we examine Baker's Brill--Noether existence conjecture for special divisors. For $g\leq 5$ and $\rho(g,r,d)$ non-negative, every graph of genus $g$ is shown to admit a divisor of rank $r$ and degree at most $d$. Moreover, the conjecture is shown to hold in rank $1$ for a number of families of highly connected combinatorial types of graphs of arbitrarily high genus. In the relevant genera, our arguments give the first combinatorial proof of the Brill--Noether existence theorem for metric graphs, giving a partial answer to a related question of Baker.