Graphs of gonality three

Abstract

In 2013, Chan classified all metric hyperelliptic graphs, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge-and vertex-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three.

Gonality sequences of graphs

“…(2) If D(v a ) = 0, choose v i ∈ supp(D), and consider D − (v i ) − (v a ). The same argument from case (1) shows that the debt on v a cannot be eliminated, a contradiction. We can now prove that any reasonable pair of first two gonalities is achieved by some graph.…”

“…For example, one can use other invariants of the graph, such as edge-connectivity. The following lower bound on gonality is stated in [14] and proved in [1].…”

“…Finally, for genus 5, the gonality sequence beginning with first gonality 3 is achieved by the genus 5 version of the graph G used in Lemma 4.1. The final gonality sequence beginning with first gonality 4 is achieved by the graph illustrated in Table4.1 (this graph has first gonality 4 by[1, Proposition 4.5]…”

Divisorial and geometric gonality of higher-rank tropical curves

“…The geometric gonality of Γ is ggon(Γ) = min{deg(ϕ) : T a tree, ϕ : Γ → T a harmonic, non-degenerate morphism} Remark 2.15. This definition is based on the notion of geometric gonality of combinatorial graphs in [1]. Alternatively, geometric gonality may be defined as the minimum degree of an indexed harmonic morphism onto a tree ([5, Def 2.1] and [14, Def.…”