Let $G$ be a graph in which each vertex has degree divisible by four. Prove or disprove that there's a subgraph $G^\prime$ on the same vertex set such that the degrees in $G^\prime$ are half the original degrees.
The previous part asked to do the same for degrees divisible by 2, and there, a triangle works because the sum of degrees is even. But here, I'm clueless... How to translate the divisibility information into anything geometric?
Pick $H$ a connected component of $G$, we must only show that a subgraph of $H$ with the desired properties exists.
The number of edges in $H$ is half of the sum of the degrees in $H$, since this sum is a multiple of $4$, the number of edges in $H$ is even.
Since all of the degrees in $H$ are even there exists an eulerian circuit $e_1,e_2,\dots e_{2m}$ of $H$, the subgraph we want is the one that contains the edges with even index.