143 research outputs found
A proof of the Ryser-Brualdi-Stein conjecture for large even
A Latin square of order is an by grid filled using symbols so
that each symbol appears exactly once in each row and column. A transversal in
a Latin square is a collection of cells which share no symbol, row or column.
The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every
Latin square of order contains a transversal with cells, and a
transversal with cells if is odd. Keevash, Pokrovskiy, Sudakov and
Yepremyan recently improved the long-standing best known bounds towards this
conjecture by showing that every Latin square of order has a transversal
with cells. Here, we show, for sufficiently large ,
that every Latin square of order has a transversal with cells.
We also apply our methods to show that, for sufficiently large , every
Steiner triple system of order has a matching containing at least
edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and
Yepremyan, who found such matchings with edges, and
proves a conjecture of Brouwer from 1981 for large .Comment: 71 pages, 13 figure
A general approach to transversal versions of Dirac-type theorems
Given a collection of hypergraphs with the same
vertex set, an -edge graph is a transversal if
there is a bijection such that for
each . How large does the minimum degree of each need to be so
that necessarily contains a copy of that is a transversal?
Each in the collection could be the same hypergraph, hence the minimum
degree of each needs to be large enough to ensure that .
Since its general introduction by Joos and Kim [Bull. Lond. Math. Soc., 2020,
52(3):498-504], a growing body of work has shown that in many cases this lower
bound is tight. In this paper, we give a unified approach to this problem by
providing a widely applicable sufficient condition for this lower bound to be
asymptotically tight. This is general enough to recover many previous results
in the area and obtain novel transversal variants of several classical
Dirac-type results for (powers of) Hamilton cycles. For example, we derive that
any collection of graphs on an -vertex set, each with minimum degree at
least , contains a transversal copy of the -th power of a
Hamilton cycle. This can be viewed as a rainbow version of the P\'osa-Seymour
conjecture.Comment: 21 pages, 4 figures; final version as accepted for publication in the
Bulletin of the London Mathematical Societ
Paths and cycles in graphs and hypergraphs
In this thesis we present new results in graph and hypergraph theory all of which feature paths or cycles.
A -uniform tight cycle is a -uniform hypergraph on vertices with a cyclic ordering of its vertices such that the edges are all -sets of consecutive vertices in the ordering.
We consider a generalisation of Lehel's Conjecture, which states that every 2-edge-coloured complete graph can be partitioned into two cycles of distinct colour, to -uniform hypergraphs and prove results in the 4- and 5-uniform case.
For a -uniform hypergraph~, the Ramsey number is the smallest integer such that any 2-edge-colouring of the complete -uniform hypergraph on vertices contains a monochromatic copy of . We determine the Ramsey number for 4-uniform tight cycles asymptotically in the case where the length of the cycle is divisible by 4, by showing that = (5+(1)).
We prove a resilience result for tight Hamiltonicity in random hypergraphs. More precisely, we show that for any >0 and 3 asymptotically almost surely, every subgraph of the binomial random -uniform hypergraph in which all -sets are contained in at least edges has a tight Hamilton cycle.
A random graph model on a host graph is said to be 1-independent if for every pair of vertex-disjoint subsets of , the state of edges (absent or present) in is independent of the state of edges in . We show that = 4 - 2 is the critical probability such that every 1-independent graph model on where each edge is present with probability at least contains an infinite path
The existence of subspace designs
We prove the existence of subspace designs with any given parameters,
provided that the dimension of the underlying space is sufficiently large in
terms of the other parameters of the design and satisfies the obvious necessary
divisibility conditions. This settles an open problem from the 1970s. Moreover,
we also obtain an approximate formula for the number of such designs.Comment: 61 page
A general approach to transversal versions of Dirac-type theorems
Given a collection of hypergraphs =(1,...,) with the same vertex set, an -edge graph ⊂∪∈[] is atransversal if there is a bijection ∶()→[] such that ∈(()) for each ∈(). How large does the minimum degree of each need to be so that necessarily contains a copy of that is a transversal? Each in the collection could be the same hypergraph,hence the minimum degree of each needs to be large enough to ensure that ⊆. Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020)498–504), a growing body of work has shown that inmany cases this lower bound is tight. In this paper, wegive a unified approach to this problem by providinga widely applicable sufficient condition for this lowerbound to be asymptotically tight. This is general enoughto recover many previous results in the area and obtainnovel transversal variants of several classical Dirac-typeresults for (powers of) Hamilton cycles. For example, wederive that any collection of graphs on an -vertex set, each with minimum degree at least (∕( + 1) +(1)), contains a transversal copy of the th power of a Hamilton cycle. This can be viewed as a rainbow versionof the Pósa–Seymour conjecture
Universality for transversal Hamilton cycles
Let be a graph collection on a common
vertex set of size such that for every
. We show that contains every Hamilton cycle pattern.
That is, for every map there is a Hamilton cycle whose
-th edge lies in .Comment: 18 page
Smoothed Analysis of the Koml\'os Conjecture: Rademacher Noise
The {\em discrepancy} of a matrix is given by
. An outstanding conjecture, attributed to Koml\'os,
stipulates that , whenever is a Koml\'os matrix,
that is, whenever every column of lies within the unit sphere. Our main
result asserts that holds
asymptotically almost surely, whenever is
Koml\'os, is a Rademacher random matrix, , and . We conjecture that suffices for the same assertion to hold. The factor
normalising is essentially best possible.Comment: For version 2, the bound on the discrepancy is improve
Transversals via regularity
Given graphs all on the same vertex set and a graph with
, a copy of is transversal or rainbow if it contains at most
one edge from each . When , such a copy contains exactly one edge
from each . We study the case when is spanning and explore how the
regularity blow-up method, that has been so successful in the uncoloured
setting, can be used to find transversals. We provide the analogues of the
tools required to apply this method in the transversal setting. Our main result
is a blow-up lemma for transversals that applies to separable bounded degree
graphs .
Our proofs use weak regularity in the -uniform hypergraph whose edges are
those where is an edge in the graph . We apply our lemma to
give a large class of spanning -uniform linear hypergraphs such that any
sufficiently large uniformly dense -vertex -uniform hypergraph with
minimum vertex degree contains as a subhypergraph. This
extends work of Lenz, Mubayi and Mycroft
On rainbow thresholds
Resolving a recent problem of Bell, Frieze, and Marbach, we establish the
threshold result of Frankston--Kahn--Narayanan--Park in the rainbow setting.Comment: 10 page
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