EXTENDING POTOČNIK AND ŠAJNA'S CONDITIONS ON VERTEX-TRANSITIVE SELF-COMPEMENTARY k-HYPERGRAPHS

Abstract : Let l be a positive integer, k = 2l or k = 2l + 1 and let n be a positive integer with $n \equiv 1$ (mod $2^{l+1}$). Potocnik and Sajna showed that if there exists a vertex-transitive self-complementary k-hypergraph of order n, then for every prime p we have $p^{n_{(p)}} \equiv 1 \pmod {2^{l+1}}$ (where $n_{(p)}$ denotes the largest integer $i$ for which $p^i$ divides $n$). Here we extend their result to any integer k and a larger class of integers n

Tournaments, 4-uniform hypergraphs, and an exact extremal result

We consider $4$-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of $5$ vertices spans either $0$ or exactly $2$ 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

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

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

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

Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer n and a prime p, let n(p)=max{i:pidividesn}. In this paper, we present a construction for vertex-transitive self-complementary k-uniform hypergraphs of order n for each integer n such that pn(p)≡1(mod2ℓ+1) for every prime p, where ℓ=max{k(2),(k−1)(2)}, and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k=2ℓ or k=2ℓ+1 due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary k-hypergraphs which have prime order p in the case where k=2ℓ or k=2ℓ+1 and p≡1(mod2ℓ+1), and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order p≡1(mod4), up to isomorphism.University of Winnipeghttps://www.sciencedirect.com/science/article/pii/S0012365X09004051?via%3DihubThis is an author-produced, peer-reviewed article that has been accepted for publication in Discrete Mathematics, but has not been copy-edited