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Chip-Firing Games on Banana Trees

Marchelle Beougher, Nila Cibu, Kexin Ding, Steven DiSilvio, Kristin Heysse, Sasha Kononova, Chan Lee, Ralph Morrison, Krish Singal

TL;DR

This work investigates chip-firing and divisor theory on banana-tree multigraphs, focusing on the gonality and related invariants. It introduces a recursive, memoized framework that yields a polynomial-time $O(n^3)$ algorithm for the gonality of banana paths, and it supplies explicit formulas for banana stars, alongside scramble-number and screewidth relationships. The authors show gonality can increase by an arbitrary amount under a single edge deletion, and they prove the Brill-Noether bound holds for all banana trees, with only finitely many equality cases. Complementary monotone/unimodal analyses and computational results illuminate the landscape of gonalities in these tree-structured graphs and validate conjectures within this graph class.

Abstract

We study chip-firing games on multigraphs whose underlying simple graphs are trees, paths, and stars, denoted as banana trees, paths, and stars respectively. We present a polynomial time algorithm to compute the divisorial gonality of banana paths, and give combinatorial formulas for the related invariants of scramble number and screewidth for any banana tree. Furthermore, we leverage banana paths to show that gonality can increase or decrease by an arbitrary amount upon deletion of a single edge, even when the resulting graph is connected. Lastly, we study banana trees and Brill-Noether theory to prove that the gonality conjecture holds for all banana trees.

Chip-Firing Games on Banana Trees

TL;DR

This work investigates chip-firing and divisor theory on banana-tree multigraphs, focusing on the gonality and related invariants. It introduces a recursive, memoized framework that yields a polynomial-time algorithm for the gonality of banana paths, and it supplies explicit formulas for banana stars, alongside scramble-number and screewidth relationships. The authors show gonality can increase by an arbitrary amount under a single edge deletion, and they prove the Brill-Noether bound holds for all banana trees, with only finitely many equality cases. Complementary monotone/unimodal analyses and computational results illuminate the landscape of gonalities in these tree-structured graphs and validate conjectures within this graph class.

Abstract

We study chip-firing games on multigraphs whose underlying simple graphs are trees, paths, and stars, denoted as banana trees, paths, and stars respectively. We present a polynomial time algorithm to compute the divisorial gonality of banana paths, and give combinatorial formulas for the related invariants of scramble number and screewidth for any banana tree. Furthermore, we leverage banana paths to show that gonality can increase or decrease by an arbitrary amount upon deletion of a single edge, even when the resulting graph is connected. Lastly, we study banana trees and Brill-Noether theory to prove that the gonality conjecture holds for all banana trees.

Paper Structure

This paper contains 6 sections, 15 theorems, 33 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. Computing gonality and other invariants of banana trees
  4. Monotone and unimodal banana graphs
  5. Banana trees and Brill-Noether theory
  6. Computational results

Key Result

Theorem 1.1

The gonality of a banana path on $n+1$ vertices can be computed in $O(n^3)$ time.

Figures (8)

  • Figure 1: A banana tree on the left, and its underlying simple graph on the right
  • Figure 2: The resulting divisors from from firing the $v_1$ followed by firing $v_0$.
  • Figure 3: A scramble of order $3$ on a graph.
  • Figure 4: A graph with screewidth at most $3$, as demonstrated by a tree-cut decomposition.
  • Figure 5: The banana path $B_{(5, 4, 2, 3, 3, 2)}$, with $7$ vertices and $6$ edge bunches
  • ...and 3 more figures

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Theorem 2.1: Theorem 1.1 in new_lower_bound
  • Theorem 2.2: Theorem 1.1 in screewidth-og
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Lemma 2.3
  • ...and 20 more