The Brill–Noether gonality conjecture for finite graphs

Let $G$ be a connected loopless multigraph, and let $g:= |E(G)| - |V(G)| + 1$ denote its cyclomatic number. Brill–Noether gonality conjecture. Then

$$\operatorname{dgon}(G) \leq \left\lfloor\frac{g + 3}{2}\right\rfloor.$$

This is the $r=1$ case of the Brill–Noether conjecture for finite graphs, an analogue of the corresponding result for algebraic curves. The paper focuses on this case; the supplied text does not establish its resolution.