Let $T_1=(V, E_1), T_2=(V,E_2)$ be trees on the same set of vertices, and let $G=(V,E_1 \cup E_2)$ be the graph resulting from the union of the two trees. Prove that there exist a vertex with degree of at most $3$.
My attempt, suppose that for all $v\in G, d(v)>3$, so the easy cases are if $G$ is a tree or has leafs.
Trying to show that there is a vertex such that $d(v)=2$, since both $T_{1,2}$ are trees, that means they must have leafs, if we have a connection between a leaf in $T_1$ to any point in $T_2$, then that leaf has a degree of $2$ in $G$.
I can't find a way prove that there's a vertex with a degree $3$ though...
Also, I just saw this: Prove that in two trees with same vertices there exists a vertex that the sum of its' degree in both trees is maximum $3$
I don't understand why $\sum d(v)\ge 4n$? shouldn't that be $2n$?
In the answer to the earlier question TheNotMe is using the hypothesis that $d_G(v)>3$ for each $v\in V$. Thus, $d_G(v)\ge 4$ for each $v\in V$, and $\sum_{v\in V}d_G(v)\ge 4n$, since $|V|=n$.