Uniform scrambles on graphs

TL;DR

This paper introduces the concept of scramble on graphs, explores its properties and applications in calculating gonality, and examines the computational complexity of related problems, providing new tools for graph analysis.

Contribution

It generalizes the notion of brambles to scrambles, develops methods to compute scramble number and gonality for various graphs, and analyzes the complexity of the egg-cut number.

Findings

Calculated scramble number and gonality for hypercube graphs.

Established bounds for large families of graphs.

Analyzed the computational complexity of the egg-cut number.

Abstract

A scramble on a connected multigraph is a collection of connected subgraphs that generalizes the notion of a bramble. The maximum order of a scramble, called the scramble number of a graph, was recently developed as a tool for lower bounding divisorial gonality. We present results on the scramble of all connected subgraphs with a fixed number of vertices, using these to calculate scramble number and gonality both for large families of graphs, and for specific examples like the $4$- and $5$-dimensional hypercube graphs. We also study the computational complexity of the egg-cut number of a scramble.