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.
Spanning trees in the square of pseudorandom graphs
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
We show that for every $\Delta\in\mathbb N$, there exists a constant $C$ such that if $G$ is an $(n,d,\lambda)$-graph with $d/\lambda\ge C$ and $d$ is large enough, then $G^2$ contains every $n$-vertex tree with maximum degree bounded by $\Delta$. This answers a question of Krivelevich.
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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.