183 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
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
Counting spanning subgraphs in dense hypergraphs
We give a simple method to estimate the number of distinct copies of some
classes of spanning subgraphs in hypergraphs with high minimum degree. In
particular, for each and , we show that every
-graph on vertices with minimum codegree at least
\cases{\left(\dfrac{1}{2}+o(1)\right)n & if $(k-\ell)\mid k$,\\ & \\
\left(\dfrac{1}{\lceil \frac{k}{k-\ell}\rceil(k-\ell)}+o(1)\right)n & if
$(k-\ell)\nmid k$,}
contains Hamilton -cycles as long as
. When this gives a simple proof of a result
of Glock, Gould, Joos, K\"uhn and Osthus, while, when this
gives a weaker count than that given by Ferber, Hardiman and Mond or, when
, by Ferber, Krivelevich and Sudakov, but one that holds for an
asymptotically optimal minimum codegree bound
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
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
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
Counting oriented trees in digraphs with large minimum semidegree
Let be an oriented tree on vertices with maximum degree at most
. If is a digraph on vertices with minimum
semidegree , then contains as a
spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they
only require maximum degree ). This generalizes the corresponding
result by Koml\'os, S\'ark\"ozy and Szemer\'edi for graphs. We investigate the
natural question how many copies of the digraph contains. Our main
result states that every such contains at least
copies of , which is optimal. This implies
the analogous result in the undirected case.Comment: 24 page
Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor
We prove that for and an absolute constant , if and is a random subset of where
each is included in independently with probability for
each , then asymptotically almost surely there is an order-
Latin square in which the entry in the th row and th column lies in
. The problem of determining the threshold probability for the
existence of an order- Latin square was raised independently by Johansson,
by Luria and Simkin, and by Casselgren and H{\"a}ggkvist; our result provides
an upper bound which is tight up to a factor of and strengthens the
bound recently obtained by Sah, Sawhney, and Simkin. We also prove analogous
results for Steiner triple systems and -factorizations of complete graphs,
and moreover, we show that each of these thresholds is at most the threshold
for the existence of a -factorization of a nearly complete regular bipartite
graph.Comment: 32 pages, 1 figure. Final version, to appear in Transactions of the
AM
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