Scramble number and tree-cut decompositions

TL;DR

This paper introduces the screewidth invariant for graphs, explores its properties, and establishes it as an upper bound for scramble number, with implications for chip-firing games and divisorial gonality.

Contribution

It defines the screewidth, analyzes its properties, and investigates its relationship with scramble number and divisorial gonality.

Findings

Screewidth bounds scramble number from above.

Screewidth and scramble number are not always equal.

Conjectures relate screewidth to divisorial gonality.

Abstract

The scramble number of a graph is an invariant recently developed to study chip-firing games and divisorial gonality. In this paper we introduce the screewidth of a graph, based on a variation of the existing literature on tree-cut decompositions. We prove that this invariant serves as an upper bound on scramble number, though they are not always equal. We study properties of screewidth, and present results and conjectures on its connection to divisorial gonality.