Triangle‐factors in pseudorandom graphs

Abstract

We show that if the second eigenvalue λ of a d‐regular graph G on n∈3Z vertices is at most εd2/false(nprefixlognfalse), for a small constant ε>0, then G contains a triangle‐factor. The bound on λ is at most an O(logn) factor away from the best possible one: Krivelevich, Sudakov and Szabó, extending a construction of Alon, showed that for every function d=d(n) such that Ω(n2/3)⩽d⩽n and infinitely many n∈N, there exists a d‐regular triangle‐free graph G with Θ(n) vertices and λ=Ω(d2/n).

Clique-factors in sparse pseudorandom graphs

“…Thus by Theorem 2.1, any (n, d, λ)-graph is (λ, d n)-bijumbled. Nenadov's result [15] asserts that for p ∈ (0, 1], any (εp t−1 n log n, p)-bijumbled graph of minimum degree at least pn 2 contains a K t -factor if ε = ε t > 0 is sufficiently small and t n. It is straightforward to generalize Theorem 1.2 to K t -factors and (εp t n, p)-bijumbled graphs with minimum degree pn 2 and ε = ε t > 0 sufficiently small.…”

“…A condition for arbitrary 2-factors. In his concluding remarks, Nenadov [15] raises the question whether the condition λ = o(p 2 n log n) is sufficient to force any (λ, p)-bijumbled graph G of minimum degree Ω(pn) to contain any given 2-factor, i.e., any n-vertex 2-regular graph. Since any 2-factor consists of vertex-disjoint cycles whose lengths add up to n, the problem is thus to find any given collection of such cycles in G. We will return to this question elsewhere [7], with a positive answer to Nenadov's question.…”

“…The first nontrivial result for this study was the result in [3] showing that λ ≤ cd 3t 2 n 1−3t 2 for some c = c(t) > 0 is sufficient to guarantee t-powers of Hamilton cycles (and thus K t+1 -factors when (t + 1) n). The aforementioned result of Nenadov [15] generalizes to K t -factors [15], giving the condition λ ≤ cd t−1 (n t−2 log n) for some constant c = c(t) > 0. The purpose of this note is to present a proof that, under the condition λ ≤ cd t n t−1 for some suitable c = c(t) > 0, any (n, d, λ)-graph contains a K t -factor.…”

“…In fact, the result in [3] is that this condition on λ is enough to guarantee the appearance of squares of Hamilton cycles (the square of a Hamilton cycle is obtained by connecting distinct vertices at distance at most 2 in the cycle). Another piece of evidence in support of Conjecture 1.1 is a result in [8,9] that states that, under the condition λ ≤ (1 600)d 2 n, any (n, d, λ)-graph G with n sufficiently large contains a 'near-perfect K 3 -factor'; in fact, G contains a family of vertexdisjoint copies of K 3 covering all but at most n 647 648 vertices of G. Very recently, Nenadov [15] proved that λ ≤ cd 2 (n log n) for some constant c > 0 yields a K 3 -factor. 3 With regard to K t -factors for general t ∈ N, Krivelevich, Sudakov and Szabó [12] remark that the condition λ ≤ cd t−1 n t−2 yields, for appropriate c = c(t), a fractional K t -factor.…”

Factors and loose Hamilton cycles in sparse pseudo-random hypergraphs

“…Thus, while bounds exist for many classes of graphs F (see e.g. [3]), only few are known to be (essentially) best possible: triangles, odd cycles, perfect matchings, Hamilton cycles and triangle-factors [4,5,29,38].(Linear) pseudo-random hypergraphs. In this paper we investigate the corresponding question for hypergraphs.…”