# Graph $G$ has k vertices of odd degree, prove there is $\dfrac{k}{2}$ non-closed trails that cover all the edges of the graph exactly once.

A graph $G$ has k vertices of odd degree, prove that there is $\dfrac{k}{2}$ non-closed trails on the graph that cover all the edges of the graph exactly once.

Here is my proof:

The graph has $k$ vertices, represent them by the sequence $a_1$,$a_2,$...,$a_k$

Add the $\dfrac{k}{2}$ edges into the graph to connect the $k$ vertices.

{$a_1,a_2$},{$a_3,a_4$},{$a_5,a_6$},...{$a_{k-1},a_k$}, where each pair of curly bracket represents an edge between the two vertices.

Now all $k$ vertices have even degree, which means there is a Eulerian trail T.

Then erase the $\dfrac{k}{2}$ edges, which makes the Eulerian trail break into $\dfrac{k}{2}$ separate trails that traverse all the edges in the original graph $G$ exactly once.

I'm making the assumption that the original graph $G$ is connected. Is this proof rigorous enough? Am I missing anything that might break my proof?Thanks for any help.

Your proof looks good. It might be worth mentioning why $k$ must be even at the start. You do need the assumption that $G$ is connected (or at least can be made connected by adding in $k/2$ extra edges in that way), because otherwise the result would not be true - you need $k/2$ trails just to deal with all the odd-degree vertices, so if there is another nontrivial component that doesn't have any of the odd-degree vertices in then you can't reach it without adding an extra trail.
• About your suggestion mentioning why $k$ must be even. There is a theorem that sum of degrees of all vertices should be even. Since each vertex has an odd degree, for the total sum to be even, $k$ needs to be even. Jan 23, 2017 at 18:21