622 research outputs found
Hipergráfok = Hypergraphs
A projekt célkitűzéseit sikerült megvalósÃtani. A négy év során több mint száz kiváló eredmény született, amibÅ‘l eddig 84 dolgozat jelent meg a téma legkiválóbb folyóirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. Számos régóta fennálló sejtést bebizonyÃtottunk, egész régi nyitott problémát megoldottunk hipergráfokkal kapcsolatban illetve kapcsolódó területeken. A problémák némelyike sok éve, olykor több évtizede nyitott volt. Nem egy közvetlen kutatási eredmény, de szintén bizonyos értékmérÅ‘, hogy a résztvevÅ‘k egyike a Norvég Királyi Akadémia tagja lett és elnyerte a Steele dÃjat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize
Generalisation : graphs and colourings
The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe
Euler tours in hypergraphs
We show that a quasirandom -uniform hypergraph has a tight Euler tour
subject to the necessary condition that divides all vertex degrees. The
case when is complete confirms a conjecture of Chung, Diaconis and Graham
from 1989 on the existence of universal cycles for the -subsets of an
-set.Comment: version accepted for publication in Combinatoric
Recommended from our members
Hypergraph models of metabolism
In this paper, we employ a directed hypergraph model to investigate the extent to which environmental variability influences the set of available biochemical reactions within a living cell. Such an approach avoids the limitations of the usual complex network formalism by allowing for the multilateral relationships (i.e. connections involving more than two nodes) that naturally occur within many biological processes. More specifically, we extend the concept of network reciprocity to complex hyper-networks, thus enabling us to characterise a network in terms of the existence of mutual hyper-connections, which may be considered a proxy for metabolic network complexity. To demonstrate these ideas, we study 115 metabolic hyper-networks of bacteria, each of which can be classified into one of 6 increasingly varied habitats. In particular, we found that reciprocity increases significantly with increased environmental variability, supporting the view that organism adaptability leads to increased complexities in the resultant biochemical networks
Hamilton cycles in hypergraphs below the Dirac threshold
We establish a precise characterisation of -uniform hypergraphs with
minimum codegree close to which contain a Hamilton -cycle. As an
immediate corollary we identify the exact Dirac threshold for Hamilton
-cycles in -uniform hypergraphs. Moreover, by derandomising the proof of
our characterisation we provide a polynomial-time algorithm which, given a
-uniform hypergraph with minimum codegree close to , either finds a
Hamilton -cycle in or provides a certificate that no such cycle exists.
This surprising result stands in contrast to the graph setting, in which below
the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We
also consider tight Hamilton cycles in -uniform hypergraphs for , giving a series of reductions to show that it is NP-hard to determine
whether a -uniform hypergraph with minimum degree contains a tight Hamilton cycle. It is therefore
unlikely that a similar characterisation can be obtained for tight Hamilton
cycles.Comment: v2: minor revisions in response to reviewer comments, most pseudocode
and details of the polynomial time reduction moved to the appendix which will
not appear in the printed version of the paper. To appear in Journal of
Combinatorial Theory, Series
Bounding the Number of Hyperedges in Friendship -Hypergraphs
For , an -uniform hypergraph is called a friendship
-hypergraph if every set of vertices has a unique 'friend' - that
is, there exists a unique vertex with the property that for each
subset of size , the set is a hyperedge.
We show that for , the number of hyperedges in a friendship
-hypergraph is at least , and we
characterise those hypergraphs which achieve this bound. This generalises a
result given by Li and van Rees in the case when .
We also obtain a new upper bound on the number of hyperedges in a friendship
-hypergraph, which improves on a known bound given by Li, van Rees, Seo and
Singhi when .Comment: 14 page
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