Graph Genera and Minors

My undergraduate honors thesis, advised by François Clément, is about how graphs sit on surfaces — embeddings, rotation systems, genus, and forbidden minors.

I start with the history: where the crossing number $\mathrm{cr}(G)$ came from, Heawood’s map-coloring conjecture, and how rotation systems became the standard way to encode an embedding. From there I work through PAGE, the genus algorithm I developed with Alexander Metzger, with worked examples of graphs whose genus it settled. The last part turns to the forbidden toroidal minors — the finite list of obstructions that decides which graphs fit on the torus — covering what is known and the open problems I find most interesting, like the genus of the Balaban $(3,11)$-cage.