754 research outputs found
Self-Complementary Hypergraphs
In this thesis, we survey the current research into self-complementary hypergraphs,
and present several new results.
We characterize the cycle type of the permutations on n elements with order equal
to a power of 2 which are k-complementing. The k-complementing permutations map
the edges of a k-uniform hypergraph to the edges of its complement. This yields a test
to determine whether a finite permutation is a k-complementing permutation, and
an algorithm for generating all self-complementary k-uniform hypergraphs of order
n, up to isomorphism, for feasible n. We also obtain an alternative description of
the known necessary and sufficient conditions on the order of a self-complementary
k-uniform hypergraph in terms of the binary representation of k.
We examine the orders of t-subset-regular self-complementary uniform hyper-
graphs. These form examples of large sets of two isomorphic t-designs. We restate
the known necessary conditions on the order of these structures in terms of the binary
representation of the rank k, and we construct 1-subset-regular self-complementary
uniform hypergraphs to prove that these necessary conditions are sufficient for all
ranks k in the case where t = 1.
We construct vertex transitive self-complementary k-hypergraphs of order n for
all integers n which satisfy the known necessary conditions due to Potocnik and Sajna, and consequently prove that these necessary conditions are also sufficient. We
also generalize Potocnik and Sajna's necessary conditions on the order of a vertex transitive self-complementary uniform hypergraph for certain ranks k to give neces-
sary conditions on the order of these structures when they are t-fold-transitive. In
addition, we use Burnside's characterization of transitive groups of prime degree to
determine the group of automorphisms and antimorphisms of certain vertex transitive self-complementary k-uniform hypergraphs of prime order, and we present an
algorithm to generate all such hypergraphs.
Finally, we examine the orders of self-complementary non-uniform hypergraphs,
including the cases where these structures are t-subset-regular or t-fold-transitive. We find necessary conditions on the order of these structures, and we present constructions to show that in certain cases these necessary conditions are sufficient.University of OttawaDoctor of Philosophy in Mathematic
Hipergráfok = Hypergraphs
A projekt cĂ©lkitűzĂ©seit sikerĂĽlt megvalĂłsĂtani. A nĂ©gy Ă©v során több mint száz kiválĂł eredmĂ©ny szĂĽletett, amibĹ‘l eddig 84 dolgozat jelent meg a tĂ©ma legkiválĂłbb folyĂłirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. Számos rĂ©gĂłta fennállĂł sejtĂ©st bebizonyĂtottunk, egĂ©sz rĂ©gi nyitott problĂ©mát megoldottunk hipergráfokkal kapcsolatban illetve kapcsolĂłdĂł terĂĽleteken. A problĂ©mák nĂ©melyike sok Ă©ve, olykor több Ă©vtizede nyitott volt. Nem egy közvetlen kutatási eredmĂ©ny, de szintĂ©n bizonyos Ă©rtĂ©kmĂ©rĹ‘, hogy a rĂ©sztvevĹ‘k egyike a NorvĂ©g Királyi 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
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
Tournaments, 4-uniform hypergraphs, and an exact extremal result
We consider -uniform hypergraphs with the maximum number of hyperedges
subject to the condition that every set of vertices spans either or
exactly hyperedges and give a construction, using quadratic residues, for
an infinite family of such hypergraphs with the maximum number of hyperedges.
Baber has previously given an asymptotically best-possible result using random
tournaments. We give a connection between Baber's result and our construction
via Paley tournaments and investigate a `switching' operation on tournaments
that preserves hypergraphs arising from this construction.Comment: 23 pages, 6 figure
Toward a Formal Semantics for Autonomic Components
Autonomic management can improve the QoS provided by parallel/ distributed
applications. Within the CoreGRID Component Model, the autonomic management is
tailored to the automatic - monitoring-driven - alteration of the component
assembly and, therefore, is defined as the effect of (distributed) management
code. This work yields a semantics based on hypergraph rewriting suitable to
model the dynamic evolution and non-functional aspects of Service Oriented
Architectures and component-based autonomic applications. In this regard, our
main goal is to provide a formal description of adaptation operations that are
typically only informally specified. We contend that our approach makes easier
to raise the level of abstraction of management code in autonomic and adaptive
applications.Comment: 11 pages + cover pag
Erdos-Hajnal-type theorems in hypergraphs
The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free,
that is, it does not contain an induced copy of a given graph H, then it must
contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0
depends only on the graph H. Except for a few special cases, this conjecture
remains wide open. However, it is known that a H-free graph must contain a
complete or empty bipartite graph with parts of polynomial size. We prove an
analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform
hypergraph on n vertices is H-free, for any given H, then it must contain a
complete or empty tripartite subgraph with parts of order c(log n)^{1/2 +
d(H)}, where d(H) > 0 depends only on H. This improves on the bound of c(log
n)^{1/2}, which holds in all 3-uniform hypergraphs, and, up to the value of the
constant d(H), is best possible. We also prove that, for k > 3, no analogue of
the standard Erdos-Hajnal conjecture can hold in k-uniform hypergraphs. That
is, there are k-uniform hypergraphs H and sequences of H-free hypergraphs which
do not contain cliques or independent sets of size appreciably larger than one
would normally expect.Comment: 15 page
Balanced walls for random groups
We study a random group G in the Gromov density model and its Cayley complex
X. For density < 5/24 we define walls in X that give rise to a nontrivial
action of G on a CAT(0) cube complex. This extends a result of Ollivier and
Wise, whose walls could be used only for density < 1/5. The strategy employed
might be potentially extended in future to all densities < 1/4.Comment: 18 pages, 2 figures. v2: Minor improvements, final versio
Hypergraph Learning with Line Expansion
Previous hypergraph expansions are solely carried out on either vertex level
or hyperedge level, thereby missing the symmetric nature of data co-occurrence,
and resulting in information loss. To address the problem, this paper treats
vertices and hyperedges equally and proposes a new hypergraph formulation named
the \emph{line expansion (LE)} for hypergraphs learning. The new expansion
bijectively induces a homogeneous structure from the hypergraph by treating
vertex-hyperedge pairs as "line nodes". By reducing the hypergraph to a simple
graph, the proposed \emph{line expansion} makes existing graph learning
algorithms compatible with the higher-order structure and has been proven as a
unifying framework for various hypergraph expansions. We evaluate the proposed
line expansion on five hypergraph datasets, the results show that our method
beats SOTA baselines by a significant margin
Generalized Kneser coloring theorems with combinatorial proofs
The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the
Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also
relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its
extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof
of the Kneser conjecture.
Here we provide a hypergraph coloring theorem, with a combinatorial proof,
which has as special cases the Kneser conjecture as well as its extensions and
generalization by (hyper)graph coloring theorems of Dol'nikov,
Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of
Schrijver's theorem.Comment: 19 pages, 4 figure
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