Hamdi Dërvodeli

Construction of tropical morphisms from tropical modifications of nonhyperelliptic genus 3 metric graphs with tree gonality 3 to metric trees

In this article, we look into the tree gonality of genus $3$ metric graphs $Γ$ which is defined as the minimum of degrees of all tropical morphisms from any tropical modification of $Γ$ to any metric tree. It is denoted by tgon$(Γ)$ and is at most $3$. We define hyperelliptic metric graphs in terms of tropical morphisms and tree gonality. Let $Γ$ be a genus $3$ metric graph with tgon$(Γ) = 3$ which is not hyperelliptic. In this paper, for such metric graphs $Γ$, we construct a tropical modification $Γ'$ of $Γ$, a metric tree $T$ and a tropical map $φ:Γ' \to T$ of degree $3$.