# graph theory proof : number of vertices of odd degree indicates edge property

I am working on this proof and sort of stuck on how to start:

Prove that if G is a connected graph with exactly 4 vertices of odd degree, there exist two trails in G such that each edge is in exactly one trail.

If anybody can give me some hints or help on how to start it! Thanks

• we learned about euler path in class and if a graph has more than 2 odd degree vertices it can not have a euler path – Vera Feb 1 '18 at 22:08

Hint: Let your four odd-degree vertices be called $a,b,c,d$. Consider the multigraph formed by taking your original graph and adding an edge between $a$ and $b$ and an edge between $c$ and $d$.