## § Euler tours

#### § Tucker's proof: undirected graph with all even degree has an euler tour

I find this proof much more intuitive because it's extremely clear where the even condition is used.
1. For each vertex $v$, arbitrarily pair edges incident at $v$ to get chains u --- v --- w.
2. Connect chains to get cycles.
3. Find a spanning tree of cycles to get an euler tour, where we have an edge between two cycles if they share a common vertex.
It's super clear why we need all vertices to be even degree; You can't pair up vertices otherwise!