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).