Gonality and edge subdivisions

Definitions and mathematical statement

For a finite connected loopless graph $G$, a divisor is an integer vector $D\in\mathbb Z^{V(G)}$, of degree $\sum_vD(v)$. It is effective if every coordinate is nonnegative. Two divisors are equivalent if their difference is $L_Gz$ for an integer vector $z$, where $(L_Gz)(v)=\sum_{vw\in E(G)}(z(v)-z(w))$ counts edge multiplicities. An effective divisor has rank at least one if $D-\mathbf1_v$ is equivalent to an effective divisor for every vertex $v$. The divisorial gonality $\operatorname{gon}(G)$ is the minimum degree of such a divisor.

Let $G^{(k)}$ replace every edge by a path of $k+1$ edges, inserting $k$ new vertices per edge. Determine how $\operatorname{gon}(G^{(k)})$ depends on $G$ and $k$, including sharp bounds relative to $\operatorname{gon}(G)$ and conditions for equality or strict decrease. One specific remaining question asks whether strict decrease can occur when $k+1>1$ is odd.

Uniform subdivision does not always preserve divisorial gonality: the cited counterexamples already give a decrease when $k=1$. Divisors on the subdivided graph may put chips on newly inserted vertices, and equivalence uses its new Laplacian. This discrete problem does not silently replace vertex-supported divisors by arbitrary points of a metric graph.

Short English statement

How does subdividing every edge equally affect discrete divisorial gonality, and can an odd number of replacement segments cause a decrease?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000040 (AMR-029-0040), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Egres Open - List of open problems, Open-problem page: Gonality and edge subdivisions. Original source indexed by AMR-029-0040. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
• S3 EGRES Open, Gonality and edge subdivisions. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/Gonality_and_edge_subdivisions.
• S4 J. van Dobben de Bruyn, H. Smit and M. van der Wegen, Discrete and metric divisorial gonality can be different, especially the final open questions. https://arxiv.org/abs/2106.12568.