Embedding nearly-spanning bounded degree trees

Abstract

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, λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolute values) has large enough spectral gap D/λ as a function of d and , then G has a copy of every tree T as above.

Sharp threshold for the appearance of certain spanning trees in random graphs

“…Let X A be the random variable that counts the number of edges of G with both endpoints in A. Then X A ∼ Bin a 2 , p and thus E(X A ) = a 2 p. Let E 3 denote the event "there exists a set A ⊆ [n], of size 1 ≤ a ≤ n(log log n) 2 log n , such that e G (A) > a log n log log n ". Using the bound (1) we get…”

“…While this was not stated explicitly in [2], one can observe that a very similar approach works for another class of bounded degree spanning trees. To this end, we need to introduce the notion of a bare path.…”

“…In particular, Alon, Sudakov and the second author proved in [2] that for given ε > 0 and integer d there exists C = C(d, ε) > 0 such that a.a.s. 1 the random graph G(n, p) with p = C/n admits a copy of a tree T on (1 − ε)n vertices with maximum degree at most d (in fact, it was proved in [2] that such a random graph contains a.a.s. a copy of every such tree).…”

“…(We repeatedly write "a given tree" to stress the order of quantifiers -first a tree is given and only then a random graph G ∼ G(n, p) is exposed. Our aim is then to find a copy of that particular tree T in G. 2 Stronger, universality-type statements, which assert that G(n, p) a.a.s. simultaneously contains every tree T from a given class, are usually much harder to obtain.)…”

“…simultaneously contains every tree T from a given class, are usually much harder to obtain.) The authors of [2] have observed that if a tree T has at least αn leaves for some constant α > 0, then a.a.s. G(n, C log n/n) contains a copy of T for some sufficiently large C = C(α) > 0.…”