377 research outputs found
Fractional clique decompositions of dense graphs
For each , we show that any graph with minimum degree at least
has a fractional -decomposition. This improves the best
previous bounds on the minimum degree required to guarantee a fractional
-decomposition given by Dukes (for small ) and Barber, K\"uhn, Lo,
Montgomery and Osthus (for large ), giving the first bound that is tight up
to the constant multiple of (seen, for example, by considering Tur\'an
graphs).
In combination with work by Glock, K\"uhn, Lo, Montgomery and Osthus, this
shows that, for any graph with chromatic number , and any
, any sufficiently large graph with minimum degree at least
has, subject to some further simple necessary
divisibility conditions, an (exact) -decomposition.Comment: 15 pages, 1 figure, submitte
Monochromatic Clique Decompositions of Graphs
Let be a graph whose edges are coloured with colours, and be a -tuple of graphs. A monochromatic -decomposition of is a partition of the edge set of such that each
part is either a single edge or forms a monochromatic copy of in colour
, for some . Let be the smallest
number , such that, for every order- graph and every
-edge-colouring, there is a monochromatic -decomposition with at
most elements. Extending the previous results of Liu and Sousa
["Monochromatic -decompositions of graphs", Journal of Graph Theory},
76:89--100, 2014], we solve this problem when each graph in is a
clique and is sufficiently large.Comment: 14 pages; to appear in J Graph Theor
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
The existence of designs via iterative absorption: hypergraph -designs for arbitrary
We solve the existence problem for -designs for arbitrary -uniform
hypergraphs~. This implies that given any -uniform hypergraph~, the
trivially necessary divisibility conditions are sufficient to guarantee a
decomposition of any sufficiently large complete -uniform hypergraph into
edge-disjoint copies of~, which answers a question asked e.g.~by Keevash.
The graph case was proved by Wilson in 1975 and forms one of the
cornerstones of design theory. The case when~ is complete corresponds to the
existence of block designs, a problem going back to the 19th century, which was
recently settled by Keevash. In particular, our argument provides a new proof
of the existence of block designs, based on iterative absorption (which employs
purely probabilistic and combinatorial methods).
Our main result concerns decompositions of hypergraphs whose clique
distribution fulfills certain regularity constraints. Our argument allows us to
employ a `regularity boosting' process which frequently enables us to satisfy
these constraints even if the clique distribution of the original hypergraph
does not satisfy them. This enables us to go significantly beyond the setting
of quasirandom hypergraphs considered by Keevash. In particular, we obtain a
resilience version and a decomposition result for hypergraphs of large minimum
degree.Comment: This version combines the two manuscripts `The existence of designs
via iterative absorption' (arXiv:1611.06827v1) and the subsequent `Hypergraph
F-designs for arbitrary F' (arXiv:1706.01800) into a single paper, which will
appear in the Memoirs of the AM
Edge-decompositions of graphs with high minimum degree
A fundamental theorem of Wilson states that, for every graph , every
sufficiently large -divisible clique has an -decomposition. Here a graph
is -divisible if divides and the greatest common divisor
of the degrees of divides the greatest common divisor of the degrees of
, and has an -decomposition if the edges of can be covered by
edge-disjoint copies of . We extend this result to graphs which are
allowed to be far from complete. In particular, together with a result of
Dross, our results imply that every sufficiently large -divisible graph of
minimum degree at least has a -decomposition. This
significantly improves previous results towards the long-standing conjecture of
Nash-Williams that every sufficiently large -divisible graph with minimum
degree at least has a -decomposition. We also obtain the
asymptotically correct minimum degree thresholds of for the
existence of a -decomposition, and of for the existence of a
-decomposition, where . Our main contribution is a
general `iterative absorption' method which turns an approximate or fractional
decomposition into an exact one. In particular, our results imply that in order
to prove an asymptotic version of Nash-Williams' conjecture, it suffices to
show that every -divisible graph with minimum degree at least
has an approximate -decomposition,Comment: 41 pages. This version includes some minor corrections, updates and
improvement
A bandwidth theorem for approximate decompositions
We provide a degree condition on a regular -vertex graph which ensures
the existence of a near optimal packing of any family of bounded
degree -vertex -chromatic separable graphs into . In general, this
degree condition is best possible.
Here a graph is separable if it has a sublinear separator whose removal
results in a set of components of sublinear size. Equivalently, the
separability condition can be replaced by that of having small bandwidth. Thus
our result can be viewed as a version of the bandwidth theorem of B\"ottcher,
Schacht and Taraz in the setting of approximate decompositions.
More precisely, let be the infimum over all
ensuring an approximate -decomposition of any sufficiently large regular
-vertex graph of degree at least . Now suppose that is an
-vertex graph which is close to -regular for some and suppose that is a sequence of bounded
degree -vertex -chromatic separable graphs with . We show that there is an edge-disjoint packing of
into .
If the are bipartite, then is sufficient. In
particular, this yields an approximate version of the tree packing conjecture
in the setting of regular host graphs of high degree. Similarly, our result
implies approximate versions of the Oberwolfach problem, the Alspach problem
and the existence of resolvable designs in the setting of regular host graphs
of high degree.Comment: Final version, to appear in the Proceedings of the London
Mathematical Societ
- …