A note on self-complementary 4-uniform hypergraphs

We prove that a permutation $\theta$ is complementing permutation for a $4$-uniform hypergraph if and only if one of the following cases is satisfied: (i) the length of every cycle of $\theta$ is a multiple of $8$, (ii) $\theta$ has $1$, $2$ or $3$ fixed points, and all other cycles have length a multiple of $8$, (iii) $\theta$ has $1$ cycle of length $2$, and all other cycles have length a multiple of $8$, (iv) $\theta$ has $1$ fixed point, $1$ cycle of length $2$, and all other cycles have length a multiple of $8$, (v) $\theta$ has $1$ cycle of length $3$, and all other cycles have length a multiple of $8$. Moreover, we present algorithms for generating every possible $3$ and $4$-uniform self-complementary hypergraphs

Self-complementing permutations of k-uniform hypergraphs

Graphs and Algorithm

Ample simplicial complexes

Motivated by potential applications in network theory, engineering and computer science, we study $r$-ample simplicial complexes. These complexes can be viewed as finite approximations to the Rado complex which has a remarkable property of {\it indestructibility,} in the sense that removing any finite number of its simplexes leaves a complex isomorphic to itself. We prove that an $r$-ample simplicial complex is simply connected and $2$-connected for $r$ large. The number $n$ of vertexes of an $r$-ample simplicial complex satisfies $\exp(\Omega(\frac{2^r}{\sqrt{r}}))$. We use the probabilistic method to establish the existence of $r$-ample simplicial complexes with $n$ vertexes for any $n>r 2^r 2^{2^r}$. Finally, we introduce the iterated Paley simplicial complexes, which are explicitly constructed $r$-ample simplicial complexes with nearly optimal number of vertexes

Hypergraphs, existential closure, and related problems

In this thesis, we present results from multiple projects with the theme of extending results from graphs to hypergraphs. We first discuss the existential closure property in graphs, a property that is known to hold for most graphs but in practice, examples of these graphs are hard to find. Specifically, we focus on finding necessary conditions for the existence of existentially closed line graphs and line graphs of hypergraphs. We then present constructions for generating infinite families of existentially closed line graphs. Interestingly, when restricting ourselves to existentially closed planar line graphs, we find that there are only finitely many such graphs. Next, we consider the notion of an existentially closed hypergraph, a novel concept that retains many of the necessary properties of an existentially closed graph. Again, we present constructions for generating infinitely many existentially closed hypergraphs. These constructions use combinatorial designs as the key ingredients, adding to the expansive list of applications of combinatorial designs. Finally, we extend a classical result of Mader concerning the edge-connectivity of vertextransitive graphs to linear uniform vertex-transitive hypergraphs. Additionally, we show that if either the linear or uniform properties are absent, then we can generate infinite families of vertex-transitive hypergraphs that do not satisfy the conclusion of the generalised theorem

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

From Large to In nite Random Simplicial Complexes.

PhD ThesesRandom simplicial complexes are a natural higher dimensional generalisation to the models of random graphs from Erd}os and R enyi of the early 60s. Now any topological question one may like to ask raises a question in probability - i.e. what is the chance this topological property occurs? Several models of random simplicial complexes have been intensely studied since the early 00s. This thesis introduces and studies two general models of random simplicial complexes that includes many well-studied models as a special case. We study their connectivity and Betti numbers, prove a satisfying duality relation between the two models, and use this to get a range of results for free in the case where all probability parameters involved are uniformly bounded. We also investigate what happens when we move to in nite dimensional random complexes and obtain a simplicial generalisation of the Rado graph, that is we show the surprising result that (under a large range of parameters) every in nite random simplicial complexes is isomorphic to a given countable complex X with probability one. We show that this X is in fact homeomorphic to the countably in nite ball. Finally, we look at and construct nite approximations to this complex X, and study their topological properties

Second Generation General System Theory: Perspectives in Philosophy and Approaches in Complex Systems

Following the classical work of Norbert Wiener, Ross Ashby, Ludwig von Bertalanffy and many others, the concept of System has been elaborated in different disciplinary fields, allowing interdisciplinary approaches in areas such as Physics, Biology, Chemistry, Cognitive Science, Economics, Engineering, Social Sciences, Mathematics, Medicine, Artificial Intelligence, and Philosophy. The new challenge of Complexity and Emergence has made the concept of System even more relevant to the study of problems with high contextuality. This Special Issue focuses on the nature of new problems arising from the study and modelling of complexity, their eventual common aspects, properties and approaches—already partially considered by different disciplines—as well as focusing on new, possibly unitary, theoretical frameworks. This Special Issue aims to introduce fresh impetus into systems research when the possible detection and correction of mistakes require the development of new knowledge. This book contains contributions presenting new approaches and results, problems and proposals. The context is an interdisciplinary framework dealing, in order, with electronic engineering problems; the problem of the observer; transdisciplinarity; problems of organised complexity; theoretical incompleteness; design of digital systems in a user-centred way; reaction networks as a framework for systems modelling; emergence of a stable system in reaction networks; emergence at the fundamental systems level; behavioural realization of memoryless functions

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..