44 research outputs found
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
Minimum number of additive tuples in groups of prime order
For a prime number and a sequence of integers , let be the minimum number of
-tuples with
, over subsets of
sizes respectively. An elegant argument of Lev (independently
rediscovered by Samotij and Sudakov) shows that there exists an extremal
configuration with all sets being intervals of appropriate length, and
that the same conclusion also holds for the related problem, reposed by Bajnok,
when and , provided is not equal 1 modulo
. By applying basic Fourier analysis, we show for Bajnok's problem that if
and are fixed while tends to
infinity, then the extremal configuration alternates between at least two
affine non-equivalent sets.Comment: This version is the same as the published version except for
modifications to reflect Reference [5], that was brought to our attention
after publicatio
The exact minimum number of triangles in graphs of given order and size
What is the minimum number of triangles in a graph of given order and size?
Motivated by earlier results of Mantel and Tur\'an, Rademacher solved the first
non-trivial case of this problem in 1941. The problem was revived by Erd\H{o}s
in 1955; it is now known as the Erd\H{o}s-Rademacher problem. After attracting
much attention, it was solved asymptotically in a major breakthrough by
Razborov in 2008. In this paper, we provide an exact solution for all large
graphs whose edge density is bounded away from~, which in this range
confirms a conjecture of Lov\'asz and Simonovits from 1975. Furthermore, we
give a description of the extremal graphs.Comment: Published in Forum of Mathematics, Pi, Volume 8, e8 (2020
The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques
Let be a sequence of natural numbers. For a
graph , let denote the number of colourings of the edges
of with colours such that, for every , the
edges of colour contain no clique of order . Write
to denote the maximum of over all graphs on vertices.
This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it
has been solved only for a very small number of non-trivial cases.
We prove that, for every and , there is a complete
multipartite graph on vertices with . Also, for every we construct a finite
optimisation problem whose maximum is equal to the limit of as tends to infinity. Our final result is a
stability theorem for complete multipartite graphs , describing the
asymptotic structure of such with in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So
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
On degree sequences forcing the square of a Hamilton cycle
A famous conjecture of P\'osa from 1962 asserts that every graph on
vertices and with minimum degree at least contains the square of a
Hamilton cycle. The conjecture was proven for large graphs in 1996 by Koml\'os,
S\'ark\"ozy and Szemer\'edi. In this paper we prove a degree sequence version
of P\'osa's conjecture: Given any , every graph of sufficiently
large order contains the square of a Hamilton cycle if its degree sequence
satisfies for all . The degree sequence condition here is asymptotically best possible. Our
approach uses a hybrid of the Regularity-Blow-up method and the
Connecting-Absorbing method.Comment: 52 pages, 5 figures, to appear in SIAM J. Discrete Mat
The generalised Oberwolfach problem
We prove that any quasirandom dense large graph in which all degrees are
equal and even can be decomposed into any given collection of two-factors
(2-regular spanning subgraphs). A special case of this result gives a new
solution to the Oberwolfach problem.Comment: 32 pages, 4 figure
The bandwidth theorem for locally dense graphs
The Bandwidth theorem of B\"ottcher, Schacht and Taraz gives a condition on
the minimum degree of an -vertex graph that ensures contains every
-chromatic graph on vertices of bounded degree and of bandwidth
, thereby proving a conjecture of Bollob\'as and Koml\'os. In this paper
we prove a version of the Bandwidth theorem for locally dense graphs. Indeed,
we prove that every locally dense -vertex graph with contains as a subgraph any given (spanning) with bounded
maximum degree and sublinear bandwidth.Comment: 35 pages. Author accepted version, to appear in Forum of Mathematics,
Sigm