A central topic of research in graph theory is various measures of a graph's connectivity. One measure of connectivity has to do with the size of brambles. A bramble is a collection of connected subsets of a graph's vertices with the property that every pair of subsets touch. For example, the picture below (stolen from wikipedia) depicts a bramble in a 3x3 grid graph. Brambles are related to other important graph invariants, including some that are motivated by algebraic geometry.
In this project, we will be studying a related, recently defined, set of objects known as scrambles. Every bramble is a scramble but not every scramble is a bramble. In particular, we will be looking for examples of graphs that have larger scrambles than brambles. We expect such examples to be abundant, and to have applications to the geometry of algebraic curves.
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