ABSTRACT:We prove that for fixed integer D and positive reals α and γ , there exists a constant C 0 such that for all p satisfying p(n) ≥ C 0 /n, the random graph G(n, p) asymptotically almost surely contains a copy of every tree with maximum degree at most D and at most (1 − α)n vertices, even after we delete a (1/2 − γ )-fraction of the edges incident to each vertex. The proof uses Szemerédi's regularity lemma for sparse graphs and a bipartite variant of the theorem of Friedman and Pippenger on embedding bounded degree trees into expanding graphs.

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large n:(i) [1-factorization conjecture] Suppose that n is even and D ≥ 2⌈n/4⌉ − 1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ ′ (G) = D. (ii) [Hamilton decomposition conjecture] Suppose that D ≥ ⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint Hamilton cycles in a graph of given minimum degree. According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer questions of Nash-Williams from 1970. The above bounds are best possible. In the current paper, we show the following: suppose that G is close to a complete balanced bipartite graph or to the union of two cliques of equal size. If we are given a suitable set of path systems which cover a set of 'exceptional' vertices and edges of G, then we can extend these path systems into an approximate decomposition of G into Hamilton cycles (or perfect matchings if appropriate).

A remarkable result of Friedman and Pippenger gives a sufficient condition on the expansion properties of a graph to contain all small trees with bounded maximum degree. Haxell showed that under slightly stronger assumptions on the expansion rate, their technique allows one to find arbitrarily large trees with bounded maximum degree. Using a slightly weaker version of Haxell's result we prove that a certain family of expanding graphs, which includes very sparse random graphs and regular graphs with large enough spectral gap, contains all almost spanning bounded degree trees. This improves two recent tree-embedding results of Alon, Krivelevich and Sudakov.

Let G be a simple graph on n vertices. A conjecture of Bollobás and Eldridge [5] asserts that if δ(G) ≥ kn−1 k+1 then G contains any n vertex graph H with ∆(H) = k. We prove a strengthened version of this conjecture for bipartite, bounded degree H, for sufficiently large n. This is the first result on this conjecture for expander graphs of arbitrary (but bounded) degree. An important tool for the proof is a new version of the Blow-up Lemma.

Let G1 and G2 be simple graphs on n vertices. If there are edge-disjoint copies of G1 and G2 in Kn, then we say there is a packing of G1 and G2. A conjecture of Bollobás and Eldridge [5] asserts that if (∆(G1) + 1)(∆(G2) + 1) ≤ n + 1 then there is a packing of G1 and G2. We prove this conjecture when ∆(G1) = 3, for sufficiently large n.