A tight condition for triangle factors in pseudorandom graphs

An (n, d, λ)-graph is an n vertex, d-regular graph with second eigenvalue in absolute value λ. When λ is small compared to d, such graphs have pseudorandom properties and make good expander graphs. A fundamental question in the study of pseudorandom graphs is to find conditions on the parameters that guarantee the existence of a certain subgraph. A celebrated construction due to Alon gives a triangle-free (n, d, λ)-graph with d = Θ(n2/3) and λ = Θ(d2/n). This construction is optimal as having λ = o(d2/n) guarantees the existence of a triangle in an (n, d, λ)-graph. Krivelevich, Sudakov and Szabó (2004) conjectured that if n ∊ 3ℕ and λ = o(d2/n) then an (n, d, λ)-graph G in fact contains a triangle factor: vertex disjoint triangles covering the whole vertex set. This conjecture has attracted the attention of many authors but until now has evaded a full solution. In this paper1, we confirm the conjecture of Krivelevich, Sudakov and Szabó and our proof gives a randomised algorithm that finds a triangle factor. The result can be seen as a clear distinction between pseudorandom graphs and random graphs, showing that essentially the same pseudorandom condition that ensures a triangle in a graph actually guarantees a triangle factor. In fact, even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition actually guarantees that such a graph G contains every graph on n vertices with maximum degree at most 2.