A k-uniform hypergraph H = (V;E) is called self-complementary if there is a permutation σ:V → V, called self-complementing, such that for every k-subset e of V, e ∈ E if and only if σ(e) ∉ E. In other words, H is isomorphic with $H^{\text{'}}=(V;\left(\genfrac{}{}{0pt}{}{V}{k}\right)-E)$. In the present paper, for every k, (1 ≤ k ≤ n), we give a characterization of self-complementig permutations of k-uniform self-complementary hypergraphs of the order n. This characterization implies the well known results for self-complementing permutations of graphs,...

Frankl and Rödl [3] proved a strong regularity lemma for 3-uniform hypergraphs, based on the concept of δ-regularity with respect to an underlying 3-partite graph. In applications of that lemma it is often important to be able to "glue" together separate pieces of the desired subhypergraph. With this goal in mind, in this paper it is proved that every pair of typical edges of the underlying graph can be connected by a hyperpath of length at most seven. The typicality of edges is defined in terms...

We consider Stanley-Reisner rings $k[x_1,...,x_n]/I\left(\mathscr{H}\right)$ where $I\left(\mathscr{H}\right)$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize some known results about chordal graphs and study a weak form of shellability.

We investigate, using results from [[p3]], when a given lattice is isomorphic to the weak subalgebra lattice of a partial algebra of a fixed type. First, we reduce this problem to the question when hyperedges of a hypergraph can be directed to a form of directed hypergraph of a fixed type. Secondly, we show that it is enough to consider some special hypergraphs. Finally, translating these results onto the lattice language, we obtain necessary conditions for our algebraic problem, and also, we completely...

The main subject of our study are spherical (weakly spherical) graphs, i.e. connected graphs fulfilling the condition that in each interval to each vertex there is exactly one (at least one, respectively) antipodal vertex. Our analysis concerns properties of these graphs especially in connection with convexity and also with hypercube graphs. We deal e.g. with the problem under what conditions all intervals of a spherical graph induce hypercubes and find a new characterization of hypercubes: $G$ is...

Let r ≥ 2 be an integer. A real number α ∈ [0,1) is a jump for r if for any ε > 0 and any integer m ≥ r, any r-uniform graph with n > n₀(ε,m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c = c(α) > 0 does not depend on ε and m. A result of Erdös, Stone and Simonovits implies that every α ∈ [0,1) is a jump for r = 2. Erdös asked whether the same is true for r ≥ 3. Frankl and Rödl gave a negative answer by showing an infinite sequence...

A hypergraph is a sum hypergraph iff there are a finite S ⊆ IN⁺ and d̲, [d̅] ∈ IN⁺ with 1 < d̲ ≤ [d̅] such that is isomorphic to the hypergraph ${}_{d̲,\left[d̅\right]}\left(S\right)=(V,)$ where V = S and $=e\subseteq S:d̲\le \left|e\right|\le \left[d̅\right]\wedge {\sum }_{v\in e}v\in S$. For an arbitrary hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $y₁,...,y_{\sigma }\notin V$ such that $\cup y₁,...,y_{\sigma }$ is a sum hypergraph. Generalizing the graph Cₙ we obtain d-uniform hypergraphs where any d consecutive vertices of Cₙ form an edge. We determine sum numbers and investigate properties of sum labellings for this...