The tree degree sequence packing conjecture

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 sequences on the same vertex set, and write $d_v^{(i)}$ for the degree of vertex $v$ in $D_i$. They have no common leaves when, for every vertex $v$ and every $i$, $d_v^{(i)}=1$ implies $d_v^{(j)}>1$ for all $j\ne i$. Tree degree sequence packing conjecture. Any collection $D_1,D_2,\ldots,D_k$ of tree degree sequences without common leaves has edge-disjoint realizations. This extends the known cases $k=2$ and $k=3$; the paper proves the assertion for $k=4$, while the general case remains open.