2 problems

Let $T_2,T_3,\ldots,T_n$ be trees such that each $T_i$ has order $i$. A graph is $k$-chromatic if its chromatic number is $k$, and $K_{n-1,\lceil\frac{n}{2}\rceil}$ denotes the com…

A tree degree sequence is a list of positive integers whose sum is $2n-2$; a realization is a graph with exactly those vertex degrees. Let $D_1,D_2,\ldots,D_k$ be tree degree seque…