"Jordan Curve Theorem: a proof using graphs."
Jordan Curve Theorem states that every non-self-intersecting closed
curve divides the plane into two regions. Although this theorem seems obvious, Bolzano was the first to point out that proving this fact is not easy. The first proof is historically attributed to Camille Jordan in 1887, and in the last century many other proofs have been discovered. Some of the most recent ones are short and elementary. For instance, the one given by Carsten Thomassen in 1992 uses only elementary results from Graph Theory. In this talk I will sketch this proof, and mention some of the difficulties that make this theorem not so obvious.