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).
The scramble number of the KC graph 
with 
and 
is 
(Echavarria et al. 2021).
