show that if $T$ is a tree containing at least one vertex of degree $2$ then $\overline T$ is not Eulerian.
I know that every tree has at least 2 leaves, so they can't be Eulerian. So in $\overline T$, these 2 leaves will adjacent to eachother. But I'm now sure if they stay being odd vertices or not. Is there a way to link the degree of vertices of $T$ and its complement?
A leaf of T has degree $1$, and there is a vertex with degree $2$. So, the complement of T has a vertex with degree $n-2$ and a vertex with degree $n-3$, so the complement of T cannot have vertices which degrees are all even.