Catalan-many tropical morphisms to trees; Part I: Constructions
Jan Draisma, Alejandro Vargas

TL;DR
This paper establishes an upper bound on the tree gonality of genus-g metric graphs using combinatorial constructions, providing a purely tropical proof that aligns with classical algebraic geometry results.
Contribution
It offers a combinatorial constructive proof for the maximum tree gonality of genus-g graphs, avoiding algebraic geometry, and characterizes the family of morphisms for even genus.
Findings
Tree gonality is at most ⌈g/2⌉ + 1.
For even genus, the morphisms form a pure-dimensional family.
The number of morphisms matches Catalan numbers in a related setting.
Abstract
We investigate the tree gonality of a genus- metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this number is at most , a fact whose proofs so far required an algebro-geometric detour via special divisors on curves. For even genus, the tropical morphism which realizes the bound belongs to a family of tropical morphisms that is pure of dimension and that has a generically finite-to-one map onto the moduli space of genus- metric graphs. Our methods focus on the study of such families. This is part I in a series of two papers: in part I we fix the combinatorial type of the metric graph, while in part II we vary the combinatorial type and show that the number of tropical morphisms, counted with suitable multiplicities, is…
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