Hamilton cycles in pseudorandom graphs

Abstract

Finding general conditions which ensure that a graph is Hamiltonian is a central topic in graph theory. An old and well known conjecture in the area states that any d-regular n-vertex graph G whose second largest eigenvalue in absolute value λ(G) is at most d/C, for some universal constant C > 0, has a Hamilton cycle. In this paper, we obtain two main results which make substantial progress towards this problem. Firstly, we settle this conjecture in full when the degree d is at least a small power of n. Secondly, in the general case we show that λ(G) ≤ d/C(log n) 1/3 implies the existence of a Hamilton cycle, improving the 20-year old bound of d/ log 1−o(1) n of Krivelevich and Sudakov. We use in a novel way a variety of methods, such as a robust Pósa rotation-extension technique, the Friedman-Pippenger tree embedding with rollbacks and the absorbing method, combined with additional tools and ideas.Our results have several interesting applications, giving best bounds on the number of generators which guarantee the Hamiltonicity of random Cayley graphs, which is an important partial case of the well known Hamiltonicity conjecture of Lovász. They can also be used to improve a result of Alon and Bourgain on additive patterns in multiplicative subgroups.

Dirac-Type Conditions for Spanning Bounded-Degree Hypertrees

“…As pointed out by Glock, Munhá Correira, and Sudakov [9], it is not even known how to find paths of length longer than n−O( λn d ) in (n, d, λ)-graphs when d/λ ≥ C for some large C. Therefore, looking for general spanning trees in optimal pseudorandom graphs seems to be a quite challenging problem. For almost-spanning trees, however, Alon, Krivelevich, and Sudakov [1] showed that for any ∆ ∈ N and ε > 0, there exists a constant C = C(ε, ∆) such that if G is an (n, d, λ)-graph with d/λ ≥ C, then G contains a copy of each tree with maximum degree bounded by ∆ and at most (1 − ε)n vertices.…”

“…The main tool is a powerful embedding technique, sometimes called extendability methods or tree embeddings with rollbacks, which was first introduced by Friedman and Pippenger [7] in 1987 and subsequently improved by Haxell [11] in 2001. Here we will use a modern reformulation of this technique which is attributed to Glebov, Johannsen, and Krivelevich [8], and which has played a major role in the solution of several problems in the last few years (see [3,4,6,9,10,15,13,18,19] for instance). Roughly speaking, the extendability method (Lemma 3.13) says that if we are given a subgraph S i ⊂ G which is 'extendable' and G has good expansion properties, then we can extend S i by adding a leaf e i with one of its endpoints in V (S i ) and other in V (G) \ V (S i ) so that S i + e i remains extendable.…”

“…Krivilevich and Sudakov [16] proved that d/λ ≥ C log n 1−o (1) is enough to guarantee Hamiltonicity in (n, d, λ)-graphs, and quite recently Glock, Munhá Correira, and Sudakov [9] improved this result by showing that d/λ ≥ C(log n) 1/3 is sufficient to force Hamiltonicity in (n, d, λ)-graphs. Moreover, they also proved that Conjecture 1.2 is true when d ≥ n α for some fixed α > 0.…”