Tiling Random Regular Graphs Efficiently.

We show that for every epsilon > 0 there exists a sufficiently large d(0) is an element of N such that for every d >= d(0), whp the random d-regular graph G(n, d) contains a T-factor for every tree T on at most (1 - epsilon) d/ log d vertices. This is best possible since, for large enough integer d, whp G(n, d) does not contain a (1+epsilon)(d)/log d-star-factor. Our method gives a randomised algorithm which whp finds said T-factor and whose expected running time is O(n(1+o(1))), as well as an efficient deterministic counterpart.