Tutte’s Theorem
Aman Agarwal
In this talk, I will present Tutte’s Theorem. This theorem extends Hall’s marriage theorem,
which I presented in the first talk, to arbitrary graphs. We will see that the notion of alternating
paths, which was used in the proof of Hall’s theorem, is also useful in proving Tutte’s theorem. I
will also discuss some implications and extensions of Tutte’s theorem.
Here’s the theorem: An undirected graph G = (V, E) has a perfect matching if and only if for
every subset U of V, the subgraph induced by V – U has at most |U| connected components
with an odd number of vertices.