  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.

Algebraic Geometry;Combinatorics