# Upper bound on the divisorial gonality of a graph

$\rm{gon}(G) \leq \frac{|E(G)|-|V(G)|}{2} + 2$, where $\rm{gon}(G)$ the denotes the divisorial gonality of graph $G$.

## Remarks

• It is a conjecture of Matthew Baker based on extensive computer calculations by Adam Tart [1].
• In the context of graph divisor theory $|E(G)|-|V(G)|+ 1$ is called as the genus of the graph and denoted by $g$. Using this notion, the conjecture can be formulated as $\rm{gon}(G) \leq \lfloor \frac{g+3}{2} \rfloor$, which is the original form of the conjecture, see Conj. 3.10. in [1].
• It is also conjectured in [1] that for each integer $g \geq 0$, there exists a graph of genus $g$ with gonality exactly $\lfloor \frac{g+3}{2} \rfloor$.
• There is also a more general version of the conjecture in [1], the so-called Brill-Noether Conjecture for Graphs: Fix integers $g, r, d \geq 0$, and set $\varrho(g, r, d) = g − (r + 1)(g − d + r)$. Then:
• (1) If $\varrho(g, r, d) \geq 0$, then every graph of genus $g$ has a divisor $D$ with $\rm{rank}(D) = r$ and $\deg(D) \leq d$.
• (2) If $\varrho(g, r, d) \lt 0$, then there exists a graph of genus $g$ for which there is no divisor D with $\rm{rank}(D) = r$ and $\deg(D) \leq d$.

### References

1. M. Baker, Specialization of linear systems from curves to graphs, Algebra and Number Theory (2008) DOI link, Author Link