11 problems

Ringel's conjecture. Every tree with $n+1$ vertices packs $2n+1$ times into the complete graph $K_{2n+1}$.

Balanced tree decomposition conjecture. Any tree $T_{n,n}$ decomposes $K_{2n-1,2n-1}$.

Hobbs–Bourgeois–Kasiraj conjecture. Any family of trees $T_1,\dots,T_n$ packs into $K_{\lceil n/2\rceil,n-1}$.

Graham–Häggkvist complete-bipartite conjecture. Any tree of order $n+1$ decomposes the complete bipartite graph $K_{n,n}$.

Böttcher–Hladký–Piguet–Taraz conjecture. Each family of trees of individual orders at most $n+1$ and total number of edges at most $\binom{2n+1}{2}$ packs into $K_{2n+1}$.

Let $k,n\in\mathbb{N}\setminus\{1\}$. Let $\mathcal{T}$ be a family of $k$-trees, where a $k$-tree is the recursively defined $k$-uniform tree obtained from one edge by repeatedly…

Let $k,n\in\mathbb{N}\setminus\{1\}$. Let $T$ be a $k$-tree, meaning a $k$-uniform tree defined recursively by starting with one edge and successively adding a vertex together with…

Let $T_1,\dotsc,T_n$ be trees, and let $v(T_i)$ denote the number of vertices of $T_i$. A collection of graphs packs into a graph $G$ if there are edge-disjoint subgraphs of $G$ is…

Average-degree tree-packing conjecture. If $G$ has average degree at least $k-1$, equivalently at least

Minimum-degree tree-packing conjecture. If $G$ satisfies

Chromatic tree-packing conjecture. If $G$ is a $k$-chromatic graph, then the set of trees $T_2,\dots,T_k$ has a packing into $G$.