A New Lower Bound on Graph Gonality

TL;DR

This paper introduces the scramble number, a new graph invariant that provides bounds for gonality and treewidth, offering insights into graph complexity beyond traditional measures.

Contribution

The paper defines the scramble number and demonstrates its effectiveness as a lower bound for gonality and an upper bound for treewidth, with computations for various graph families.

Findings

Scramble number bounds gonality from below.

Scramble number bounds treewidth from above.

Computed values for multiple graph families.

Abstract

We define a new graph invariant called the scramble number. We show that the scramble number of a graph is a lower bound for the gonality and an upper bound for the treewidth. Unlike the treewidth, the scramble number is not minor monotone, but it is subgraph monotone and invariant under refinement. We compute the scramble number and gonality of several families of graphs for which these invariants are strictly greater than the treewidth.