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Let $G=(V,U,E)$ be a bipartite graph, i.e. $V,U$ are sets of vertices s.t. $E\subseteq V\times U$.

It is also given that every vertex (both in $U$ and in $V$) has an even degree.

Prove that there exist some subset $E'\subseteq E\ $ s.t. in the graph $G'=(V,U,E')$ the degree of each vertex is exactly half of its degree in $G$.

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Hint: A graph $G$ in which every vertex has even degree can be decomposed into cycles. (That is, $E(G)$ is a disjoint union of cycles.) If $G$ is bipartite, what has to be true about those cycles?

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    Every vertex of the graph must be in an even number of cycles? and then we can take the cycles and leave half of the edges in a yes no operation, like taking a flower and taking one out and leaving the next? – 2017-02-20
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    @TrueTopologist A vertex need not be in an even number of cycles, but your idea about alternately taking and deleting edges is correct. – 2017-02-20
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    Oh thank you. So it will work too because each vertex will stay with one instead of two edges in each cycle, right? – 2017-02-20