Graham–Häggkvist conjecture for complete bipartite hosts

A graph decomposes $G$ if copies of the graph form a perfect packing of $G$, and $K_{n,n}$ denotes the complete bipartite graph with $n$ vertices in each part.

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

This follows from the stated conjecture that every tree of order $n+1$ decomposes every $n$-regular bipartite graph. The source reports approximate results and evidence, but the exact assertion remains open.