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So I understand that if we prove for the largest case we can subsequently prove for any following cases they are subgraphs with the same property, but I'm having trouble proving this largest case. A hint I was given was to consider 2k-regular graphs.

Can anyone help with understanding this proof?

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    That's a direct corollary of [this question](https://math.stackexchange.com/questions/1691770/deg-v-deg-v-le-1). – 2017-02-08
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    Or [this](https://math.stackexchange.com/questions/1533684/how-to-prove-that-every-graph-admits-an-almost-balanced-orientation) or [this](https://math.stackexchange.com/questions/396009/prove-the-edges-of-a-multigraph-may-be-oriented-such-that-the-net-degree-of-any)... – 2017-02-08

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