16,461 research outputs found
The extremal function for partial bipartite tilings
For a fixed bipartite graph H and given number c, 0<c<1, we determine the
threshold T_H(c) which guarantees that any n-vertex graph with at edge density
at least T_H(c) contains vertex-disjoint copies of H. In the
proof we use a variant of a technique developed by
Komlos~\bcolor{[Combinatorica 20 (2000), 203-218}]Comment: 10 page
On the critical pair theory in abelian groups : Beyond Chowla's Theorem
We obtain critical pair theorems for subsets S and T of an abelian group such
that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman
and a more recent result due to Rodseth and one of the authors.Comment: Submitted to Combinatorica, 23 pages, revised versio
Random strategies are nearly optimal for generalized van der Waerden Games
In a (1 : q) Maker-Breaker game, one of the central questions is to find (or at least estimate) the maximal value of q that allows Maker to win the game. Based on the ideas of Bednarska and Luczak [Bednarska, M., and T. Luczak, Biased positional games for which random strategies are nearly optimal, Combinatorica, 20 (2000), 477–488], who studied biased H-games, we prove general winning criteria for Maker and Breaker and a hypergraph generalization of their result. Furthermore, we study the biased version of a strong generalization of the van der Waerden games introduced by Beck [Beck, J., Van der Waerden and Ramsey type games, Combinatorica, 1 (1981), 103–116] and apply our criteria to determine the threshold bias of these games up to constant factor. As in the result of [Bednarska, M., and T. Luczak, Biased positional games for which random strategies are nearly optimal, Combinatorica, 20 (2000), 477–488], the random strategy for Maker is again the best known strategy.Postprint (updated version
Hipergráfok = Hypergraphs
A projekt célkitűzéseit sikerült megvalósítani. A négy év során több mint száz kiváló eredmény született, amiből 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 bebizonyítottunk, 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 értékmérő, hogy a résztvevők egyike a Norvég Királyi Akadémia tagja lett és elnyerte a Steele díjat. | 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
Transversals in -Uniform Hypergraphs
Let be a -regular -uniform hypergraph on vertices. The
transversal number of is the minimum number of vertices that
intersect every edge. Lai and Chang [J. Combin. Theory Ser. B 50 (1990),
129--133] proved that . Thomass\'{e} and Yeo [Combinatorica
27 (2007), 473--487] improved this bound and showed that .
We provide a further improvement and prove that , which is
best possible due to a hypergraph of order eight. More generally, we show that
if is a -uniform hypergraph on vertices and edges with maximum
degree , then , which proves a known
conjecture. We show that an easy corollary of our main result is that the total
domination number of a graph on vertices with minimum degree at least~4 is
at most , which was the main result of the Thomass\'{e}-Yeo paper
[Combinatorica 27 (2007), 473--487].Comment: 41 page
The Independence Number of the Orthogonality Graph in Dimension
We determine the independence number of the orthogonality graph on
-dimensional hypercubes. This answers a question by Galliard from 2001
which is motivated by a problem in quantum information theory. Our method is a
modification of a rank argument due to Frankl who showed the analogous result
for -dimensional hypercubes, where is an odd prime.Comment: 3 pages, accepted by Combinatorica, fixed a minor typo spotted by
Peter Si
The planar Cayley graphs are effectively enumerable I: consistently planar graphs
We obtain an effective enumeration of the family of finitely generated groups
admitting a faithful, properly discontinuous action on some 2-manifold
contained in the sphere. This is achieved by introducing a type of group
presentation capturing exactly these groups.
Extending this in a companion paper, we find group presentations capturing
the planar finitely generated Cayley graphs. Thus we obtain an effective
enumeration of these Cayley graphs, yielding in particular an affirmative
answer to a question of Droms et al.Comment: To appear in Combinatorica. The second half of the previous version
is arXiv:1901.0034
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