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

Abstract

In an influential 2008 paper, Baker proposed a number of conjectures relating the Brill-Noether theory of algebraic curves with a divisor theory on finite graphs. In this note, we examine Baker's Brill-Noether existence conjecture for special divisors. For g ≤ 5 and ρ(g, r, d) non-negative, every graph of genus g is shown to admit a divisor of rank r and degree at most d. As further evidence, the conjecture is shown to hold in rank 1 for a number families of highly connected combinatorial types of graphs. In the relevant genera, our arguments give the first combinatorial proof of Brill-Noether existence theorem for metric graphs, giving a partial answer to a related question of Baker. Divisor theory on finite graphsThe main reference for this section is the original paper of Baker and Norine [4]. A graph G will mean a finite connected graph possibly with loops and multiple edges. The vertex and edge sets of G will be denoted V (G) and E(G) respectively. The genus of G, denoted g(G), is defined to be g(G) := |E(G)| − |V (G)| + 1.