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
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 $d$.
What nice properties does this new multigraph have? How many vertices of odd degree exist in this multigraph? What does this imply? How can we use that to our advantage in regards to the original graph?