310 research outputs found
Colouring of Graphs with Ramsey-Type Forbidden Subgraphs
A colouring of a graph G = (V;E) is a mapping c : V ! f1; 2; : : :g such that c(u) 6= c(v) if uv 2 E; if jc(V )j k then c is a k-colouring. The Colouring problem is that of testing whether a given graph has a k-colouring for some given integer k. If a graph contains no induced subgraph isomorphic to any graph in some family H, then it is called H-free. The complexity of Colouring for H-free graphs with jHj = 1 has been completely classied. When jHj = 2, the classication is still wide open, although many partial results are known. We continue this line of research and forbid induced subgraphs fH1;H2g, where we allow H1 to have a single edge and H2 to have a single nonedge. Instead of showing only polynomial-time solvability, we prove that Colouring on such graphs is xed-parameter tractable when parameterized by jH1j + jH2j. As a byproduct, we obtain the same result both for the problem of determining a maximum independent set and for the problem of determining a maximum clique
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
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Erdos-Hajnal-type theorems in hypergraphs
The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free,
that is, it does not contain an induced copy of a given graph H, then it must
contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0
depends only on the graph H. Except for a few special cases, this conjecture
remains wide open. However, it is known that a H-free graph must contain a
complete or empty bipartite graph with parts of polynomial size. We prove an
analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform
hypergraph on n vertices is H-free, for any given H, then it must contain a
complete or empty tripartite subgraph with parts of order c(log n)^{1/2 +
d(H)}, where d(H) > 0 depends only on H. This improves on the bound of c(log
n)^{1/2}, which holds in all 3-uniform hypergraphs, and, up to the value of the
constant d(H), is best possible. We also prove that, for k > 3, no analogue of
the standard Erdos-Hajnal conjecture can hold in k-uniform hypergraphs. That
is, there are k-uniform hypergraphs H and sequences of H-free hypergraphs which
do not contain cliques or independent sets of size appreciably larger than one
would normally expect.Comment: 15 page
On the Geometric Ramsey Number of Outerplanar Graphs
We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2
outerplanar triangulations in both convex and general cases. We also prove that
the geometric Ramsey numbers of the ladder graph on vertices are bounded
by and , in the convex and general case, respectively. We
then apply similar methods to prove an upper bound on the
Ramsey number of a path with ordered vertices.Comment: 15 pages, 7 figure
Degrees in oriented hypergraphs and sparse Ramsey theory
Let be an -uniform hypergraph. When is it possible to orient the edges
of in such a way that every -set of vertices has some -degree equal
to ? (The -degrees generalise for sets of vertices what in-degree and
out-degree are for single vertices in directed graphs.) Caro and Hansberg asked
if the obvious Hall-type necessary condition is also sufficient.
Our main aim is to show that this is true for large (for given ), but
false in general. Our counterexample is based on a new technique in sparse
Ramsey theory that may be of independent interest.Comment: 20 pages, 3 figure
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