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On gonality-tight graphs

Šimun Dropuljić, Yoav Len

TL;DR

This work characterizes graphs whose second gonality exceeds the first by exactly one (gonality-tight graphs) under a stronger condition, delivering a complete description of their gonality sequence for a large family. The authors prove a structure theorem: such graphs arise inductively by gluing complete graphs to existing graphs in controlled ways, and precisely identify quasi-banana graphs as the canonical family capturing gonality-tightness when the second gonality is realized by a specific divisor. Using a pair of technical lemmas that bound gonalities upon attaching complete graphs, they derive both upper and lower bounds that tightly constrain the gonality sequence. The results bridge tropical geometry and combinatorial graph theory, and extend the understanding of how a local divisor configuration dictates global gonality behavior, with implications for the moduli of graphs carrying prescribed gonality sequences.

Abstract

We address a question posed by Fessler-Jensen-Kelsey-Owen regarding graphs whose second gonality is greater than the first by exactly 1. We answer the question affirmatively under a stronger condition, thereby characterising the entire gonality sequence for a large family of graphs. We prove a structure theorem for the graphs satisfying the condition, and show that they are all obtained via an inductive process by gluing together complete and banana graphs under certain rules.

On gonality-tight graphs

TL;DR

This work characterizes graphs whose second gonality exceeds the first by exactly one (gonality-tight graphs) under a stronger condition, delivering a complete description of their gonality sequence for a large family. The authors prove a structure theorem: such graphs arise inductively by gluing complete graphs to existing graphs in controlled ways, and precisely identify quasi-banana graphs as the canonical family capturing gonality-tightness when the second gonality is realized by a specific divisor. Using a pair of technical lemmas that bound gonalities upon attaching complete graphs, they derive both upper and lower bounds that tightly constrain the gonality sequence. The results bridge tropical geometry and combinatorial graph theory, and extend the understanding of how a local divisor configuration dictates global gonality behavior, with implications for the moduli of graphs carrying prescribed gonality sequences.

Abstract

We address a question posed by Fessler-Jensen-Kelsey-Owen regarding graphs whose second gonality is greater than the first by exactly 1. We answer the question affirmatively under a stronger condition, thereby characterising the entire gonality sequence for a large family of graphs. We prove a structure theorem for the graphs satisfying the condition, and show that they are all obtained via an inductive process by gluing together complete and banana graphs under certain rules.

Paper Structure

This paper contains 12 sections, 13 theorems, 202 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Strategy of the proof of the main theorems
  3. Preliminaries
  4. Gonality-tight and quasi-banana graphs
  5. Restricting the structure of gonality-tight graphs
  6. Quasi-banana graphs
  7. Two technical lemmas
  8. Bounding gonalities from above
  9. Bounding gonalities from below
  10. Proof of the main theorems
  11. Proof of \ref{['mainTheorem:propertiesOfQBGraphs']}
  12. Proof of \ref{['mainTheorem:alternativeDefinitonForQBGraphs']}

Key Result

Theorem 1

A graph $G$ is a quasi-banana graph if and only if it is gonality-tight and there exists a vertex $v\in V(G)$ such that the divisor realises the second gonality.

Figures (10)

  • Figure 1: Two quasi-banana graphs of gonality $5$.
  • Figure 2: The graph $G$ from \ref{['lemma:structureMainLemma']}.
  • Figure 3: The result of starting a fire at $u_i$ with respect to $D_2-2u_i$. Parts of the graph that burn down are highlighted in red.
  • Figure 4: Graph $G$ after a fire is started at $w$ with respect to $D_2$. Parts of the graph on fire are highlighted in red.
  • Figure 5: The graph $G$ after starting a fire at $u_i$ with respect to $D_2$. Here, $\ell$ is the number of edges between vertices on fire other than $u_i$ and vertices of $G$ not on fire. Burnt regions are highlighted in red.
  • ...and 5 more figures

Theorems & Definitions (67)

  • Theorem 1
  • Theorem 2
  • Conjecture
  • Lemma 1.1
  • proof
  • Definition 2.1
  • Corollary 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • ...and 57 more