# Upper bound on the divisorial gonality of a graph

From Egres Open

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

## Remarks

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

### References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}M. Baker,*Specialization of linear systems from curves to graphs*, Algebra and Number Theory (2008) DOI link, Author Link