Latin squares via graph theory

Latin squares have been studied in Combinatorics since the early work of Euler in the 1780's. More recently, much progress has been made from the perspective of graph theory, allowing modern graph theory tools to be applied to an equivalent formulation using edge-coloured graphs. I will discuss in particular the study of transversals in Latin squares via this route, and the main extremal and probabilistic questions in the area:

i) How large a partial transversal can we find in any Latin square?

ii) How many disjoint transversals are we likely to find in a random Latin square?

Part of this talk is based on joint work with Candida Bowtell