363 research outputs found
Ramsey properties of randomly perturbed graphs: cliques and cycles
Given graphs , a graph is -Ramsey if for every
colouring of the edges of with red and blue, there is a red copy of
or a blue copy of . In this paper we investigate Ramsey questions in the
setting of randomly perturbed graphs: this is a random graph model introduced
by Bohman, Frieze and Martin in which one starts with a dense graph and then
adds a given number of random edges to it. The study of Ramsey properties of
randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in
2006; they determined how many random edges must be added to a dense graph to
ensure the resulting graph is with high probability -Ramsey (for
). They also raised the question of generalising this result to pairs
of graphs other than . We make significant progress on this
question, giving a precise solution in the case when and
where . Although we again show that one requires polynomially fewer
edges than in the purely random graph, our result shows that the problem in
this case is quite different to the -Ramsey question. Moreover, we
give bounds for the corresponding -Ramsey question; together with a
construction of Powierski this resolves the -Ramsey problem.
We also give a precise solution to the analogous question in the case when
both and are cycles. Additionally we consider the
corresponding multicolour problem. Our final result gives another
generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we
determine how many random edges must be added to a dense graph to ensure the
resulting graph is with high probability -Ramsey (for odd
and ).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil
Powierski, stated results for cliques in graphs of low positive density
separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to
appear in CP
Packing k-partite k-uniform hypergraphs
Let and be -graphs (-uniform hypergraphs); then a perfect
-packing in is a collection of vertex-disjoint copies of in
which together cover every vertex of . For any fixed let
be the minimum such that any -graph on vertices with
minimum codegree contains a perfect -packing. The
problem of determining has been widely studied for graphs (i.e.
-graphs), but little is known for . Here we determine the
asymptotic value of for all complete -partite -graphs ,
as well as a wide class of other -partite -graphs. In particular, these
results provide an asymptotic solution to a question of R\"odl and Ruci\'nski
on the value of when is a loose cycle. We also determine
asymptotically the codegree threshold needed to guarantee an -packing
covering all but a constant number of vertices of for any complete
-partite -graph .Comment: v2: Updated with minor corrections. Accepted for publication in
Journal of Combinatorial Theory, Series
Almost all optimally coloured complete graphs contain a rainbow Hamilton path
A subgraph of an edge-coloured graph is called rainbow if all of the
edges of have different colours. In 1989, Andersen conjectured that every
proper edge-colouring of admits a rainbow path of length . We show
that almost all optimal edge-colourings of admit both (i) a rainbow
Hamilton path and (ii) a rainbow cycle using all of the colours. This result
demonstrates that Andersen's Conjecture holds for almost all optimal
edge-colourings of and answers a recent question of Ferber, Jain, and
Sudakov. Our result also has applications to the existence of transversals in
random symmetric Latin squares.Comment: 29 pages, 5 figure
Decompositions into isomorphic rainbow spanning trees
A subgraph of an edge-coloured graph is called rainbow if all its edges have
distinct colours. Our main result implies that, given any optimal colouring of
a sufficiently large complete graph , there exists a decomposition of
into isomorphic rainbow spanning trees. This settles conjectures of
Brualdi--Hollingsworth (from 1996) and Constantine (from 2002) for large
graphs.Comment: Version accepted to appear in JCT
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
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
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