The gonality-tight graph conjecture

Let $G$ be a gonality-tight graph with first gonality $k$ for some integer $k$, and let $g$ denote its genus. Write $K_G$ for the canonical divisor of $G$, and write $D\sim D'$ when two divisors are linearly equivalent. The $r$-th gonality is denoted by $\operatorname{gon}_r(G)$.

Gonality-tight graph conjecture. The following hold:

1. The genus of $G$ is

$$g=\binom{k}{2}.$$

2. The gonality sequence is

$$\operatorname{gon}_r(G)=\begin{cases} l(k+1)-h & \text{if } r<g,\\ g+r & \text{if } r\geq g,\end{cases}$$

where $1\leq l\leq k-2$ and $0\leq h\leq l$ are uniquely determined integers satisfying

$$r=\frac{l(l+3)}{2}-h.$$

3. If $D$ realizes the second gonality, then

$$(k-2)\cdot D\sim K_G.$$

Moreover, for every integer $1\leq l\leq k-2$, all divisors realizing the $\frac{l(l+3)}{2}$-th gonality are linearly equivalent to $l\cdot D$.

The conjecture is motivated by a large number of computational examples and predicts that gonality-tight graphs have a rigid genus, gonality sequence, and linear-equivalence structure for divisors realizing selected gonalities. Its status is not established by the supplied material.