Embedding nearly-spanning bounded degree trees

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 - epsilon)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d >= 2 and 0 < epsilon < 1, there exists a constant c = c(d, E) such that a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 - epsilon)n vertices with maximum degree at most d. We also prove that if an (n, D, lambda)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most A in their absolute values) has large enough spectral gap D/lambda as a function of d and c, then G has a copy of every tree T as above.