15 problems

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

Let $G$ be a connected $d$-regular graph, let $\tau(G)$ denote the maximum number of edge-disjoint spanning trees contained in $G$, and let $\lambda_{2}(G)$ be the second-largest e…

Let $n\in\mathbb{N}$, and let $T_1,T_2,\ldots,T_n$ be trees satisfying $|V(T_i)|=i$ for each $i$. Gyárfás–Lehel's conjecture. The trees $T_1,T_2,\ldots,T_n$ can be packed into the…

Let $T_1,\dots,T_n$ be trees such that, for each $i\in[n]$, $T_i$ has $i$ vertices. Define … A decomposition of $G$ into $T_1,\dots,T_n$ is a collection of pairwise edge-disjoint c…

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 $T_1$ and $T_2$ be trees with $n$ vertices, neither of which is a star. A tight planar packing is a packing of $T_1$ and $T_2$ in the plane that meets the tightness condition f…

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

Hollingsworth's conjecture. Any family of balanced trees $T_{1,1},\dots,T_{n,n}$ packs into $K_{n,n}$.

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 $K_{n,n}$.

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…

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 $G$.