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 and nonnegative, every graph of genus is shown to admit a divisor of rank and degree at most . As further evidence, the conjecture is shown to hold in rank for a number families of highly connected combinatorial types of graphs. 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.
"A Note on Brill–Noether Existence for Graphs of Low Genus." Michigan Math. J. 67 (1) 175 - 198, March 2018. https://doi.org/10.1307/mmj/1519095622