Scramble Number

The scramble number  of a graph  is a graph invariant developed to aid in the study of gonality of graphs. The scramble number is NP-hard to compute (Echavarria et al. 2021).

The scramble number satisfies

where  is the edge connectivity and  is the vertex count of .

The scramble number is the most powerful known lower bound on the gonality of a graph and satisfies

where  is the vertex connectivity,  is the edge connectivity,  is the treewidth, and  is the gonality of  (Harp et al. 2020, Echavarria et al. 2021).

Unfortunately, the scramble number is not quite as well-behaved as treewidth (Echavarria et al. 2021).

REFERENCES

Echavarria, M.; Everett, M.; Huang, R.; Jacoby, L.; Morrison, R.; Weber, B. "On the Scramble Number of Graphs." 29 Mar 2021. https://arxiv.org/abs/2103.15253.

Harp, M.; Jackson, E.; Jensen, D.; and Speeter, N. "A New Lower Bound on Graph Gonality." 1 Jun 2020. https://arxiv.org/abs/2006.01020.