Powers of Hamilton cycles in pseudorandom graphs

Abstract

Abstract. We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (ε, p, k, ℓ)-pseudorandom if for all disjoint X and Y ⊆ V (G) with |X| ≥ εp k n and |Y | ≥ εp ℓ n we have e(X, Y ) = (1 ± ε)p|X||Y |. We prove that for all β > 0 there is an ε > 0 such that an (ε, p, 1, 2)-pseudorandom graph on n vertices with minimum degree at least βpn contains the square of a Hamilton cycle. In particular, this implies that (n, d, λ)-graphs with λ ≪ d 5/2 n −3/2 contain the square of a Hamilton cycle, and thus a triangle factor if n is a multiple of 3. This improves on a result of Krivelevich, Sudakov and Szabó [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403-426].We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions.

Triangle‐factors in pseudorandom graphs

“…Finally, it should be noted that a straightforward modification of our argument shows that every (p, εnp r−1 / log n)-bijumbled graph with minimum degree np/2 contains a K r -factor. This improves a result of Allen et al [1] which requires β = O(np 3r/2 ), though it is fair to note that they show the existence of a much richer subgraph, namely the rth power of a Hamilton cycle. It is believed that an extension of Alon's result [2] holds for larger cliques, that is, there exists a d-regular graph for d = n 1−1/(r−1) with λ(G) = Θ( √ d) which is K r -free; however, this remains a major open problem in this area.…”

“…As our main motivation comes from studying d-regular graphs with small second largest eigenvalue, for which the bijumbledness suffices, we did not pursue this. Furthermore, it would be interesting to see if β = o(np 2 / log n) is also sufficient for the existence of any 2-regular spanning subgraph, that is, any collection of cycles with the total size n. A significant strengthening of our result would be to improve a result of Allen et al [1] and show that such β ensures the square of a Hamilton cycle.…”

Finding any given 2‐factor in sparse pseudorandom graphs efficiently

“…methods, however, was first established by Rödl et al [46,47] in their study of Hamiltonicity in hypergraphs. There, the method was used to study dense hypergraphs but the methods have since been adapted to other settings (see eg [2,3,39]).…”

“…Much more is known for questions about finding one particular spanning structure in a pseudorandom graph and the most prominent spanning structures which were considered in the last fifteen years include perfect matchings, studied by Alon et al in [37], Hamilton cycles studied by Krivelevich and Sudakov [36], clique-factors [24,25,38,44] and powers of Hamilton cycles [2].…”

“…We remark that the minimum degree condition in Theorem 1.2 is weak and natural. Indeed some minimum degree condition is necessary as otherwise one could have isolated vertices and the bijumbled definition (2) guarantees that almost all vertices satisfy the minimum degree condition in any case. Finally, we mention that we do not try to optimise the running time of our algorithm and are satisfied with being able to provide a deterministic algorithm which is efficient in that it runs in polynomial time.…”