17 research outputs found
An asymptotic bound for the strong chromatic number
The strong chromatic number of a graph on
vertices is the least number with the following property: after adding isolated vertices to and taking the union with any
collection of spanning disjoint copies of in the same vertex set, the
resulting graph has a proper vertex-colouring with colours.
We show that for every and every graph on vertices with
, , which is
asymptotically best possible.Comment: Minor correction, accepted for publication in Combin. Probab. Compu
On sufficient conditions for Hamiltonicity in dense graphs
We study structural conditions in dense graphs that guarantee the existence
of vertex-spanning substructures such as Hamilton cycles. It is easy to see
that every Hamiltonian graph is connected, has a perfect fractional matching
and, excluding the bipartite case, contains an odd cycle. Our main result in
turn states that any large enough graph that robustly satisfies these
properties must already be Hamiltonian. Moreover, the same holds for embedding
powers of cycles and graphs of sublinear bandwidth subject to natural
generalisations of connectivity, matchings and odd cycles.
This solves the embedding problem that underlies multiple lines of research
on sufficient conditions for Hamiltonicity in dense graphs. As applications, we
recover and establish Bandwidth Theorems in a variety of settings including
Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type
conditions, locally dense and inseparable graphs, multipartite graphs as well
as robust expanders
Extremal results in hypergraph theory via the absorption method
The so-called "absorbing method" was first introduced in a systematic way by Rödl, Ruciński and Szemerédi in 2006, and has found many uses ever since. Speaking in a general sense, it is useful for finding spanning substructures of combinatorial structures. We establish various results of different natures, in both graph and hypergraph theory, most of them using the absorbing method:
1. We prove an asymptotically best-possible bound on the strong chromatic number with respect to the maximum degree of the graph. This establishes a weak version of a conjecture of Aharoni, Berger and Ziv.
2. We determine asymptotic minimum codegree thresholds which ensure the existence of tilings with tight cycles (of a given size) in uniform hypergraphs. Moreover, we prove results on coverings with tight cycles.
3. We show that every 2-coloured complete graph on the integers contains a monochromatic infinite path whose vertex set is sufficiently "dense" in the natural numbers. This improves results of Galvin and Erdős and of DeBiasio and McKenney
Density of monochromatic infinite paths
For any subset , we define its upper density to be
. We prove
that every -edge-colouring of the complete graph on contains a
monochromatic infinite path, whose vertex set has upper density at least . This improves on results of Erd\H{o}s and
Galvin, and of DeBiasio and McKenney.Comment: Accepted for publication in The Electronic Journal of Combinatoric
Covering and tiling hypergraphs with tight cycles
Given , we say that a -uniform hypergraph is a
tight cycle on vertices if there is a cyclic ordering of the vertices of
such that every consecutive vertices under this ordering form an
edge. We prove that if and , then every -uniform
hypergraph on vertices with minimum codegree at least has
the property that every vertex is covered by a copy of . Our result is
asymptotically best possible for infinitely many pairs of and , e.g.
when and are coprime.
A perfect -tiling is a spanning collection of vertex-disjoint copies
of . When is divisible by , the problem of determining the
minimum codegree that guarantees a perfect -tiling was solved by a
result of Mycroft. We prove that if and is not divisible
by and divides , then every -uniform hypergraph on vertices
with minimum codegree at least has a perfect
-tiling. Again our result is asymptotically best possible for infinitely
many pairs of and , e.g. when and are coprime with even.Comment: Revised version, accepted for publication in Combin. Probab. Compu
Dirac-type conditions for spanning bounded-degree hypertrees
We prove that for fixed , every -uniform hypergraph on vertices and
of minimum codegree at least contains every spanning tight -tree
of bounded vertex degree as a sub\-graph. This generalises a well-known result
of Koml\'os, S\'ark\"ozy and Szemer\'edi for graphs. Our result is
asymptotically sharp. We also prove an extension of our result to hypergraphs
that satisfy some weak quasirandomness conditions
Towards a hypergraph version of the P\'osa-Seymour conjecture
We prove that for fixed , every -uniform hypergraph on
vertices having minimum codegree at least
contains the th
power of a tight Hamilton cycle. This result may be seen as a step towards a
hypergraph version of the P\'osa--Seymour conjecture.
Moreover, we prove that the same bound on the codegree suffices for finding a
copy of every spanning hypergraph of tree-width less than which admits a
tree decomposition where every vertex is in a bounded number of bags.Comment: 22 page
Universal arrays
A word on symbols is a sequence of letters from a fixed alphabet of size
. For an integer , we say that a word is -universal if, given
an arbitrary word of length , one can obtain it by removing entries from
. It is easily seen that the minimum length of a -universal word on
symbols is exactly . We prove that almost every word of size
is -universal with high probability, where is an explicit constant
whose value is roughly . Moreover, we show that the -universality
property for uniformly chosen words exhibits a sharp threshold. Finally, by
extending techniques of Alon [Geometric and Functional Analysis 27 (2017), no.
1, 1--32], we give asymptotically tight bounds for every higher dimensional
analogue of this problem.Comment: 12 page