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# 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).

Gonality, Pebbling Number

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## 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.

## Cite this as:

Weisstein, Eric W. "Scramble Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ScrambleNumber.html