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We study Brill-Noether existence on a finite graph using methods frompolyhedral geometry and lattices. We start by formulating analogues of theBrill-Noether conjectures (both the existence and non-existence parts) for$\mathbb{R}$-divisors, i.e. divisors with real coefficients, on a graph. Wethen reformulate the Brill-Noether existence conjecture for$\mathbb{R}$-divisors on a graph in geometric terms, that we refer to as thecovering radius conjecture and we show a weak version, in support of it. Usingthis, we show an approximate version of the Brill-Noether existence conjecturefor divisors on a graph. As applications, we derive upper bounds on thegonality of a graph and its $\mathbb{R}$-divisor analogue.

Brill-Noether Existence on Graphs via $\mathbb{R}$-Divisors, Polytopes and Lattices