408 research outputs found
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
Hypergraphs and hypermatrices with symmetric spectrum
It is well known that a graph is bipartite if and only if the spectrum of its
adjacency matrix is symmetric. In the present paper, this assertion is
dissected into three separate matrix results of wider scope, which are extended
also to hypermatrices. To this end the concept of bipartiteness is generalized
by a new monotone property of cubical hypermatrices, called odd-colorable
matrices. It is shown that a nonnegative symmetric -matrix has a
symmetric spectrum if and only if is even and is odd-colorable. This
result also solves a problem of Pearson and Zhang about hypergraphs with
symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu.
Separately, similar results are obtained for the -spectram of
hypermatrices.Comment: 17 pages. Corrected proof on p. 1
An asymptotic bound for the strong chromatic number
The strong chromatic number of a graph on
vertices is the least number with the following property: after adding isolated vertices to and taking the union with any
collection of spanning disjoint copies of in the same vertex set, the
resulting graph has a proper vertex-colouring with colours.
We show that for every and every graph on vertices with
, , which is
asymptotically best possible.Comment: Minor correction, accepted for publication in Combin. Probab. Compu
Rainbow perfect matchings in r-partite graph structures
A matching M in an edge–colored (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 Lovász Lemma.Peer ReviewedPostprint (author's final draft
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
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
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
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