Upper bound on the divisorial gonality of a graph

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G=(V,E)$ be a finite connected unweighted multigraph without loops. A divisor is $D=\sum_{v\in V}D(v)v$ with $D(v)\in\mathbb Z$; it is effective if $D(v)\geq0$ for every $v$, and $\deg D=\sum_vD(v)$. For $f:V\to\mathbb Z$ define $\Delta f(v)=\sum_{e=vw}(f(v)-f(w))$, counting parallel edges separately. Divisors are linearly equivalent when their difference is $\Delta f$ for some $f$. Say $D$ has rank at least one if, for every $v\in V$, $D-v$ is equivalent to an effective divisor. The divisorial gonality $\operatorname{gon}(G)$ is the least degree of such a divisor. Set $g=|E|-|V|+1$.

Statement recorded in the supplied source.

Does every such $G$ satisfy \[\operatorname{gon}(G)\leq\frac{|E|-|V|}{2}+2=\frac{g+3}{2},\] equivalently $\operatorname{gon}(G)\leq\lfloor(g+3)/2\rfloor$? [S2]

Short English statement

Does every finite connected graph admit a divisor of rank at least one whose degree is at most $\lfloor(g+3)/2\rfloor$, where $g$ is its genus?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3000087 (AMR-029-0087). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Matthew Baker, Specialization of Linear Systems from Curves to Graphs (2007), sections 1.1, 1.3 and 3.1; Conjecture 3.10(1), pp. 12--13. https://arxiv.org/abs/math/0701075.