On the scramble number of graphs
Marino Echavarria, Max Everett, Robin Huang, Liza Jacoby, Ralph, Morrison, and Ben Weber
TL;DR
This paper proves that computing the scramble number and gonality of graphs is NP-hard, provides lower bounds for Cartesian products, and computes gonality for new graph families, advancing understanding of graph invariants.
Contribution
It establishes NP-hardness of calculating scramble number and gonality, and offers new bounds and computations for product graphs.
Findings
NP-hardness of scramble number and gonality computation
Lower bounds for scramble number of Cartesian products
Gonality calculations for new families of product graphs
Abstract
The scramble number of a graph is an invariant recently developed to aid in the study of divisorial gonality. In this paper we prove that scramble number is NP-hard to compute, also providing a proof that computing gonality is NP-hard even for simple graphs, as well as for metric graphs. We also provide general lower bounds for the scramble number of a Cartesian product of graphs, and apply these to compute gonality for many new families of product graphs.
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