329 research outputs found
Bounds on the Game Transversal Number in Hypergraphs
Let be a hypergraph with vertex set and edge set of order
\nH = |V| and size \mH = |E|. A transversal in is a subset of vertices
in that has a nonempty intersection with every edge of . A vertex hits
an edge if it belongs to that edge. The transversal game played on involves
of two players, \emph{Edge-hitter} and \emph{Staller}, who take turns choosing
a vertex from . Each vertex chosen must hit at least one edge not hit by the
vertices previously chosen. The game ends when the set of vertices chosen
becomes a transversal in . Edge-hitter wishes to minimize the number of
vertices chosen in the game, while Staller wishes to maximize it. The
\emph{game transversal number}, , of is the number of vertices
chosen when Edge-hitter starts the game and both players play optimally. We
compare the game transversal number of a hypergraph with its transversal
number, and also present an important fact concerning the monotonicity of
, that we call the Transversal Continuation Principle. It is known that
if is a hypergraph with all edges of size at least~, and is not a
-cycle, then \tau_g(H) \le \frac{4}{11}(\nH+\mH); and if is a
(loopless) graph, then \tau_g(H) \le \frac{1}{3}(\nH + \mH + 1). We prove
that if is a -uniform hypergraph, then \tau_g(H) \le \frac{5}{16}(\nH +
\mH), and if is -uniform, then \tau_g(H) \le \frac{71}{252}(\nH +
\mH).Comment: 23 pages
Rainbow perfect matchings in r-partite graph structures
A matching M in an edgeâcolored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local LovĂĄsz Lemma.Peer ReviewedPostprint (author's final draft
Decomposing 1-Sperner hypergraphs
A hypergraph is Sperner if no hyperedge contains another one. A Sperner
hypergraph is equilizable (resp., threshold) if the characteristic vectors of
its hyperedges are the (minimal) binary solutions to a linear equation (resp.,
inequality) with positive coefficients. These combinatorial notions have many
applications and are motivated by the theory of Boolean functions and integer
programming. We introduce in this paper the class of -Sperner hypergraphs,
defined by the property that for every two hyperedges the smallest of their two
set differences is of size one. We characterize this class of Sperner
hypergraphs by a decomposition theorem and derive several consequences from it.
In particular, we obtain bounds on the size of -Sperner hypergraphs and
their transversal hypergraphs, show that the characteristic vectors of the
hyperedges are linearly independent over the reals, and prove that -Sperner
hypergraphs are both threshold and equilizable. The study of -Sperner
hypergraphs is motivated also by their applications in graph theory, which we
present in a companion paper
Antichain cutsets of strongly connected posets
Rival and Zaguia showed that the antichain cutsets of a finite Boolean
lattice are exactly the level sets. We show that a similar characterization of
antichain cutsets holds for any strongly connected poset of locally finite
height. As a corollary, we get such a characterization for semimodular
lattices, supersolvable lattices, Bruhat orders, locally shellable lattices,
and many more. We also consider a generalization to strongly connected
hypergraphs having finite edges.Comment: 12 pages; v2 contains minor fixes for publicatio
Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic
The hypergraph duality problem DUAL is defined as follows: given two simple
hypergraphs and , decide whether
consists precisely of all minimal transversals of (in which case
we say that is the dual of ). This problem is
equivalent to deciding whether two given non-redundant monotone DNFs are dual.
It is known that non-DUAL, the complementary problem to DUAL, is in
, where
denotes the complexity class of all problems that after a nondeterministic
guess of bits can be decided (checked) within complexity class
. It was conjectured that non-DUAL is in . In this paper we prove this conjecture and actually
place the non-DUAL problem into the complexity class which is a subclass of . We here refer to the logtime-uniform version of
, which corresponds to , i.e., first order
logic augmented by counting quantifiers. We achieve the latter bound in two
steps. First, based on existing problem decomposition methods, we develop a new
nondeterministic algorithm for non-DUAL that requires to guess
bits. We then proceed by a logical analysis of this algorithm, allowing us to
formulate its deterministic part in . From this result, by
the well known inclusion , it follows
that DUAL belongs also to . Finally, by exploiting
the principles on which the proposed nondeterministic algorithm is based, we
devise a deterministic algorithm that, given two hypergraphs and
, computes in quadratic logspace a transversal of
missing in .Comment: Restructured the presentation in order to be the extended version of
a paper that will shortly appear in SIAM Journal on Computin
HipergrĂĄfok = Hypergraphs
A projekt cĂ©lkitƱzĂ©seit sikerĂŒlt megvalĂłsĂtani. A nĂ©gy Ă©v sorĂĄn több mint szĂĄz kivĂĄlĂł eredmĂ©ny szĂŒletett, amibĆl eddig 84 dolgozat jelent meg a tĂ©ma legkivĂĄlĂłbb folyĂłirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. SzĂĄmos rĂ©gĂłta fennĂĄllĂł sejtĂ©st bebizonyĂtottunk, egĂ©sz rĂ©gi nyitott problĂ©mĂĄt megoldottunk hipergrĂĄfokkal kapcsolatban illetve kapcsolĂłdĂł terĂŒleteken. A problĂ©mĂĄk nĂ©melyike sok Ă©ve, olykor több Ă©vtizede nyitott volt. Nem egy közvetlen kutatĂĄsi eredmĂ©ny, de szintĂ©n bizonyos Ă©rtĂ©kmĂ©rĆ, hogy a rĂ©sztvevĆk egyike a NorvĂ©g KirĂĄlyi AkadĂ©mia tagja lett Ă©s elnyerte a Steele dĂjat. | 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
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