Summary: 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 - )n vertices, in terms of the expansion properties of G. As
a result we show that for fixed d 2 and 0 < < 1, there exists a constant c = c(d, ) such that
a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 - )n vertices with
maximum degree at most d. We also prove that if an (n, D, )-graph G (i.e., a D-regular graph on
n vertices all of whose eigenvalues, except the first one, are at most in their absolute values) has
large enough spectral gap D/ as a function of d and , then G has a copy of every tree T as above.
In this paper we obtain a sufficient condition for a sparse graph G to contain a copy of every nearly-
spanning tree T of bounded maximum degree, in terms of the expansion properties of G. The restriction
on the degree of T comes naturally from the fact that we consider graphs of constant density. Two
important examples where our condition applies are random graphs and graphs with a large spectral
The random graph G(n, p) denotes the probability space whose points are graphs on a fixed set of n
vertices, where each pair of vertices forms an edge, randomly and independently, with probability p.