58 research outputs found
Equitable orientations of sparse uniform hypergraphs
Caro, West, and Yuster studied how -uniform hypergraphs can be oriented in
such a way that (generalizations of) indegree and outdegree are as close to
each other as can be hoped. They conjectured an existence result of such
orientations for sparse hypergraphs, of which we present a proof
The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques
Let be a sequence of natural numbers. For a
graph , let denote the number of colourings of the edges
of with colours such that, for every , the
edges of colour contain no clique of order . Write
to denote the maximum of over all graphs on vertices.
This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it
has been solved only for a very small number of non-trivial cases.
We prove that, for every and , there is a complete
multipartite graph on vertices with . Also, for every we construct a finite
optimisation problem whose maximum is equal to the limit of as tends to infinity. Our final result is a
stability theorem for complete multipartite graphs , describing the
asymptotic structure of such with in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So
Ramsey properties of algebraic graphs and hypergraphs
One of the central questions in Ramsey theory asks how small can be the size
of the largest clique and independent set in a graph on vertices. By the
celebrated result of Erd\H{o}s from 1947, the random graph on vertices with
edge probability , contains no clique or independent set larger than
, with high probability. Finding explicit constructions of graphs
with similar Ramsey-type properties is a famous open problem. A natural
approach is to construct such graphs using algebraic tools. Say that an
-uniform hypergraph is \emph{algebraic of complexity
} if the vertices of are elements of
for some field , and there exist polynomials
of degree at most
such that the edges of are determined by the zero-patterns of
. The aim of this paper is to show that if an algebraic graph
(or hypergraph) of complexity has good Ramsey properties, then at
least one of the parameters must be large. In 2001, R\'onyai, Babai and
Ganapathy considered the bipartite variant of the Ramsey problem and proved
that if is an algebraic graph of complexity on vertices, then
either or its complement contains a complete balanced bipartite graph of
size . We extend this result by showing that such
contains either a clique or an independent set of size
and prove similar results for algebraic hypergraphs of constant complexity. We
also obtain a polynomial regularity lemma for -uniform algebraic hypergraphs
that are defined by a single polynomial, that might be of independent interest.
Our proofs combine algebraic, geometric and combinatorial tools.Comment: 23 page
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
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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