Euler tours in hypergraphs

We show that a quasirandom $k$-uniform hypergraph $G$ has a tight Euler tour subject to the necessary condition that $k$ divides all vertex degrees. The case when $G$ is complete confirms a conjecture of Chung, Diaconis and Graham from 1989 on the existence of universal cycles for the $k$-subsets of an $n$-set.Comment: version accepted for publication in Combinatoric

Grassmann Integral Representation for Spanning Hyperforests

Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff's matrix-tree theorem. Furthermore, we show that the class of integrals describing unrooted spanning (hyper)forests is induced by a theory with an underlying OSP(1|2) supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J. Phys.

Completion and deficiency problems

Given a partial Steiner triple system (STS) of order $n$, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order $n$ with at most $r \le \varepsilon n^2$ triples, it can always be embedded into a complete STS of order $n+O(\sqrt{r})$, which is asymptotically optimal. We also obtain similar results for completions of Latin squares and other designs. This suggests a new, natural class of questions, called deficiency problems. Given a global spanning property $\mathcal{P}$ and a graph $G$, we define the deficiency of the graph $G$ with respect to the property $\mathcal{P}$ to be the smallest positive integer $t$ such that the join $G\ast K_t$ has property $\mathcal{P}$. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a $K_k$-decomposition, Hamiltonicity, having a triangle-factor and having a perfect matching in hypergraphs. The main goal of this paper is to propose a systematic study of these problems; thus several future research directions are also given

Connection and separation in hypergraphs

In this paper we study various fundamental connectivity properties of hypergraphs from a graph-theoretic perspective, with the emphasis on cut edges, cut vertices, and blocks. We prove a number of new results involving these concepts. In particular, we describe the exact relationship between the block decomposition of a hypergraph and the block decomposition of its incidence graph

Extremal problems for cycles in graphs and hypergraphs

In this thesis, we study several generalizations of Turan type problems in graphs and hypergraphs. In particular, we focus on graphs and hypergraphs without long cycles or long paths, extending famous results of Erdos and Gallai. Results include bounds on the size of such objects as well as stability theorems about the structure of extremal and almost extremal objects