134 research outputs found
Powers of Hamilton cycles in pseudorandom graphs
We study the appearance of powers of Hamilton cycles in pseudorandom graphs,
using the following comparatively weak pseudorandomness notion. A graph is
-pseudorandom if for all disjoint and with and we have
. We prove that for all there is an
such that an -pseudorandom graph on
vertices with minimum degree at least contains the square of a
Hamilton cycle. In particular, this implies that -graphs with
contain the square of a Hamilton cycle, and thus
a triangle factor if is a multiple of . This improves on a result of
Krivelevich, Sudakov and Szab\'o [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.Comment: 30 pages, 1 figur
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
The Green-Tao theorem: an exposition
The celebrated Green-Tao theorem states that the prime numbers contain
arbitrarily long arithmetic progressions. We give an exposition of the proof,
incorporating several simplifications that have been discovered since the
original paper.Comment: 26 pages, 4 figure
Approximate Hamilton decompositions of robustly expanding regular digraphs
We show that every sufficiently large r-regular digraph G which has linear
degree and is a robust outexpander has an approximate decomposition into
edge-disjoint Hamilton cycles, i.e. G contains a set of r-o(r) edge-disjoint
Hamilton cycles. Here G is a robust outexpander if for every set S which is not
too small and not too large, the `robust' outneighbourhood of S is a little
larger than S. This generalises a result of K\"uhn, Osthus and Treglown on
approximate Hamilton decompositions of dense regular oriented graphs. It also
generalises a result of Frieze and Krivelevich on approximate Hamilton
decompositions of quasirandom (di)graphs. In turn, our result is used as a tool
by K\"uhn and Osthus to prove that any sufficiently large r-regular digraph G
which has linear degree and is a robust outexpander even has a Hamilton
decomposition.Comment: Final version, published in SIAM Journal Discrete Mathematics. 44
pages, 2 figure
Finding any given 2-factor in sparse pseudorandom graphs efficiently
Given an -vertex pseudorandom graph and an -vertex graph with
maximum degree at most two, we wish to find a copy of in , i.e.\ an
embedding so that
for all . Particular instances of this problem include finding a
triangle-factor and finding a Hamilton cycle in . Here, we provide a
deterministic polynomial time algorithm that finds a given in any suitably
pseudorandom graph . The pseudorandom graphs we consider are
-bijumbled graphs of minimum degree which is a constant proportion
of the average degree, i.e.\ . A -bijumbled graph is
characterised through the discrepancy property: for any two sets of vertices and . Our condition
on bijumbledness is within a log factor from being
tight and provides a positive answer to a recent question of Nenadov.
We combine novel variants of the absorption-reservoir method, a powerful tool
from extremal graph theory and random graphs. Our approach is based on that of
Nenadov (\emph{Bulletin of the London Mathematical Society}, to appear) and on
ours (arXiv:1806.01676), together with additional ideas and simplifications.Comment: 21 page
Deterministic Approximation of Random Walks in Small Space
We give a deterministic, nearly logarithmic-space algorithm that given an undirected graph G, a positive integer r, and a set S of vertices, approximates the conductance of S in the r-step random walk on G to within a factor of 1+epsilon, where epsilon>0 is an arbitrarily small constant. More generally, our algorithm computes an epsilon-spectral approximation to the normalized Laplacian of the r-step walk.
Our algorithm combines the derandomized square graph operation [Eyal Rozenman and Salil Vadhan, 2005], which we recently used for solving Laplacian systems in nearly logarithmic space [Murtagh et al., 2017], with ideas from [Cheng et al., 2015], which gave an algorithm that is time-efficient (while ours is space-efficient) and randomized (while ours is deterministic) for the case of even r (while ours works for all r). Along the way, we provide some new results that generalize technical machinery and yield improvements over previous work. First, we obtain a nearly linear-time randomized algorithm for computing a spectral approximation to the normalized Laplacian for odd r. Second, we define and analyze a generalization of the derandomized square for irregular graphs and for sparsifying the product of two distinct graphs. As part of this generalization, we also give a strongly explicit construction of expander graphs of every size
Proof of Koml\'os's conjecture on Hamiltonian subsets
Koml\'os conjectured in 1981 that among all graphs with minimum degree at
least , the complete graph minimises the number of Hamiltonian
subsets, where a subset of vertices is Hamiltonian if it contains a spanning
cycle. We prove this conjecture when is sufficiently large. In fact we
prove a stronger result: for large , any graph with average degree at
least contains almost twice as many Hamiltonian subsets as ,
unless is isomorphic to or a certain other graph which we
specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ
Pseudorandom hypergraph matchings
A celebrated theorem of Pippenger states that any almost regular hypergraph
with small codegrees has an almost perfect matching. We show that one can find
such an almost perfect matching which is `pseudorandom', meaning that, for
instance, the matching contains as many edges from a given set of edges as
predicted by a heuristic argument.Comment: 14 page
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