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