For every fixed epsilon, with high probability the random d-regular graph contains a vertex-partition into copies of any prescribed tree of size at most (1-epsilon)d/ln d.
Crowns in pseudo-random graphs and Hamilton cycles in their squares
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abstract
A crown with $k$ spikes is an edge-disjoint union of a cycle $C$ and a matching $M$ of size $k$ such that each edge of $M$ has exactly one vertex in common with $C$. We prove that if $G$ is an $(n,d,\lambda)$-graph with $\lambda/d\le 0.001$ and $d$ is large enough, then $G$ contains a crown on $n$ vertices with $\lfloor n/2\rfloor$ spikes. As a consequence, such $G$ contains a Hamilton cycle in its square $G^2$.
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Tree tilings in random regular graphs
For every fixed epsilon, with high probability the random d-regular graph contains a vertex-partition into copies of any prescribed tree of size at most (1-epsilon)d/ln d.