77 research outputs found
Optimal path and cycle decompositions of dense quasirandom graphs
Motivated by longstanding conjectures regarding decompositions of graphs into
paths and cycles, we prove the following optimal decomposition results for
random graphs. Let be constant and let . Let be
the number of odd degree vertices in . Then a.a.s. the following hold:
(i) can be decomposed into cycles and a
matching of size .
(ii) can be decomposed into
paths.
(iii) can be decomposed into linear forests.
Each of these bounds is best possible. We actually derive (i)--(iii) from
`quasirandom' versions of our results. In that context, we also determine the
edge chromatic number of a given dense quasirandom graph of even order. For all
these results, our main tool is a result on Hamilton decompositions of robust
expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte
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
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Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combinatorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session
Path and cycle decompositions of dense graphs
We make progress on three long standing conjectures from the 1960s about path
and cycle decompositions of graphs. Gallai conjectured that any connected graph
on vertices can be decomposed into at most paths, while a conjecture of Haj\'{o}s states that any
Eulerian graph on vertices can be decomposed into at most cycles. The Erd\H{o}s-Gallai conjecture states that
any graph on vertices can be decomposed into cycles and edges.
We show that if is a sufficiently large graph on vertices with linear
minimum degree, then the following hold.
(i) can be decomposed into at most paths.
(ii) If is Eulerian, then it can be decomposed into at most
cycles.
(iii) can be decomposed into at most cycles and
edges.
If in addition satisfies a weak expansion property, we asymptotically
determine the required number of paths/cycles for each such .
(iv) can be decomposed into paths, where
is the number of odd-degree vertices of .
(v) If is Eulerian, then it can be decomposed into
cycles.
All bounds in (i)-(v) are asymptotically best possible.Comment: 48 pages, 2 figures; final version, to appear in the Journal of the
London Mathematical Societ
Minimalist designs
The iterative absorption method has recently led to major progress in the
area of (hyper-)graph decompositions. Amongst other results, a new proof of the
Existence conjecture for combinatorial designs, and some generalizations, was
obtained. Here, we illustrate the method by investigating triangle
decompositions: we give a simple proof that a triangle-divisible graph of large
minimum degree has a triangle decomposition and prove a similar result for
quasi-random host graphs.Comment: updated references, to appear in Random Structures & Algorithm
Decompositions of graphs and hypergraphs
This thesis contains various new results in the areas of design theory and edge decompositions of graphs and hypergraphs. Most notably, we give a new proof of the existence conjecture, dating back to the 19th century.
For -graphs and , an -decomposition of G is a collection of edge-disjoint copies of F in G covering all edges of . In a recent breakthrough, Keevash proved that every sufficiently large quasirandom -graph G has a -decomposition (subject to necessary divisibility conditions), thus proving the existence conjecture.
We strengthen Keevash's result in two major directions: Firstly, our main result applies to decompositions into any -graph , which generalises a fundamental theorem of Wilson to hypergraphs. Secondly, our proof framework applies beyond quasirandomness, enabling us e.g. to deduce a minimum degree version.
For graphs, we investigate the minimum degree setting further. In particular, we determine the decomposition threshold' of every bipartite graph, and show that the threshold of cliques is equal to its fractional analogue.
We also present theorems concerning optimal path and cycle decompositions of quasirandom graphs.
This thesis is based on joint work with Daniela Kuhn and Deryk Osthus, Allan Lo and Richard Montgomery
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