Emily Peters Loyola University, Chicago A fusion category is a tensor category with a finite number of simple objects and some conditions which make linear algebra work nicely (it is semisimple, k-linear and rigid). Examples of fusion categories come from quantum groups at roots of unity, and finite-depth subfactors (and that is the only mention of subfactors I will make in this talk). We are interested in describing all small trivalent categories: A trivalent category is generated by a single object and all morphisms can be draw using trivalent graphs. Why are we interested in trivalent categories? Well, first, a lot of the quantum-group and subfactor fusion categories are trivalent. Second, we have tools of linear algebra and diagrammatic combinatorics that make these calculations fun and easy. In particular, in this talk I plan to prove theorems using the method of discharging (of four-color-theorem fame).