16 research outputs found
Fractional clique decompositions of partite graphs
We give a minimum degree condition sufficient to ensure the existence of a fractionalKr-decomposition in a balancedr-partite graph (subject to some further simple necessary conditions). This generalizes the non-partite problem studied recently by Barber, Lo, Kühn, Osthus and the author, and the 3-partite fractionalK3-decomposition problem studied recently by Bowditch and Dukes. Combining our result with recent work by Barber, Kühn, Lo, Osthus and Taylor, this gives a minimum degree condition sufficient to ensure the existence of a (non-fractional)Kr-decomposition in a balancedr-partite graph (subject to the same simple necessary conditions).</jats:p
Resolution of the Oberwolfach problem
The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of
into edge-disjoint copies of a given -factor. We show that this
can be achieved for all large . We actually prove a significantly more
general result, which allows for decompositions into more general types of
factors. In particular, this also resolves the Hamilton-Waterloo problem for
large .Comment: 28 page
Threshold for Steiner triple systems
We prove that with high probability
contains a spanning Steiner triple system for ,
establishing the tight exponent for the threshold probability for existence of
a Steiner triple system. We also prove the analogous theorem for Latin squares.
Our result follows from a novel bootstrapping scheme that utilizes iterative
absorption as well as the connection between thresholds and fractional
expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.Comment: 22 pages, 1 figur
The linear system for Sudoku and a fractional completion threshold
We study a system of linear equations associated with Sudoku latin squares.
The coefficient matrix of the normal system has various symmetries arising
from Sudoku. From this, we find the eigenvalues and eigenvectors of , and
compute a generalized inverse. Then, using linear perturbation methods, we
obtain a fractional completion guarantee for sufficiently large and sparse
rectangular-box Sudoku puzzles
Clique decompositions of multipartite graphs and completion of Latin squares
Our main result essentially reduces the problem of finding an
edge-decomposition of a balanced r-partite graph of large minimum degree into
r-cliques to the problem of finding a fractional r-clique decomposition or an
approximate one. Together with very recent results of Bowditch and Dukes as
well as Montgomery on fractional decompositions into triangles and cliques
respectively, this gives the best known bounds on the minimum degree which
ensures an edge-decomposition of an r-partite graph into r-cliques (subject to
trivially necessary divisibility conditions). The case of triangles translates
into the setting of partially completed Latin squares and more generally the
case of r-cliques translates into the setting of partially completed mutually
orthogonal Latin squares.Comment: 40 pages. To appear in Journal of Combinatorial Theory, Series
Substructures in Latin squares
We prove several results about substructures in Latin squares. First, we
explain how to adapt our recent work on high-girth Steiner triple systems to
the setting of Latin squares, resolving a conjecture of Linial that there exist
Latin squares with arbitrarily high girth. As a consequence, we see that the
number of order- Latin squares with no intercalate (i.e., no
Latin subsquare) is at least . Equivalently,
, where is the number
of intercalates in a uniformly random order- Latin square.
In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the
general large-deviation problem for intercalates in random Latin squares, up to
constant factors in the exponent: for any constant we have
and for
any constant we have
.
Finally, we show that in almost all order- Latin squares, the number of
cuboctahedra (i.e., the number of pairs of possibly degenerate
subsquares with the same arrangement of symbols) is of order , which is
the minimum possible. As observed by Gowers and Long, this number can be
interpreted as measuring "how associative" the quasigroup associated with the
Latin square is.Comment: 32 pages, 1 figur
On the decomposition threshold of a given graph
We study the -decomposition threshold for a given graph .
Here an -decomposition of a graph is a collection of edge-disjoint
copies of in which together cover every edge of . (Such an
-decomposition can only exist if is -divisible, i.e. if and each vertex degree of can be expressed as a linear combination of
the vertex degrees of .)
The -decomposition threshold is the smallest value ensuring
that an -divisible graph on vertices with
has an -decomposition. Our main results imply
the following for a given graph , where is the fractional
version of and :
(i) ;
(ii) if , then
;
(iii) we determine if is bipartite.
In particular, (i) implies that . Our proof
involves further developments of the recent `iterative' absorbing approach.Comment: Final version, to appear in the Journal of Combinatorial Theory,
Series