Connected graphs without long paths

We determine the maximum number of edges in a connected graph with n vertices if it contains no path with k+1 vertices. We also determine the extremal graphs.     

On hypergraphs without significant cycles

A property of the intersection multigraph of a hypergraph is displayed. This property is then used to obtain an equality connecting the order of the hypergraph, the sizes of its edges and the number of edges of its intersection multigraph. At the end a generalization by Las Vergnas of a result of Lovász is given another proof.     

On hypergraphs without loose cycles

Recently, Mubayi and Wang showed that for $r\ge 4$ and $?\ge 3$, the number of $n$-vertex $r$-graphs that do not contain any loose cycle of length $?$ is at most $2^{O\left(n^{r?1}{(logn)}^{(r?3)∕(r?2)}\right)}$. We improve this bound to $2^{O(n^{r?1}loglogn)}$.     

The maximum size of hypergraphs without generalized 4-cycles

Let fr(n) be the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain four distinct edges A, B, C, D with A∪B=C∪D and A∩B=C∩D=∅. This problem was stated by Erd?s [P. Erd?s, Problems and results in combinatorial analysis, Congr. Numer. 19 (1977) 3-12]. It can be viewed as a generalization of the Turán problem for the 4-cycle to hypergraphs.Let . Füredi [Z. Füredi, Hypergraphs in which all disjoint pairs have distinct unions, Combinatorica 4 (1984) 161-168] observed that ?r?1 and conjectured that this is equality for every r?3. The best known upper bound ?r?3 was proved by Mubayi and Verstraëte [D. Mubayi, J. Verstraëte, A hypergraph extension of the bipartite Turán problem, J. Combin. Theory Ser. A 106 (2004) 237-253]. Here we improve this bound. Namely, we show that for every r?3, and ?₃?13/9. In particular, it follows that ?r→1 as r→∞.     

Interval hypergraphs and D-interval hypergraphs

A hypergraph $\text{H = (V,}\text{E}\text{)}$ is called an interval hypergraph if there exists a one-to-one function ? mapping the elements of V to points on the real line such that for each edge E, there is an interval I, containing the images of all elements of E, but not the images of any elements not in E₁. The difference hypergraph D(H) determined by H is formed by adding to $\text{E}$ all nonempty sets of the form E₁ ? E₁, where E₁ and E₁ are edges of HH is said to be a D-interval hypergraph if D(H) is an interval hypergraph. A forbidden subhypergraph characterization of D-interval hypergraphs is given. By relating D-interval hypergraphs to dimension theory for posets, we determine all 3-irreducible posets of length one.     

On hypergraphs without two edges intersecting in a given number of vertices

Let X be a finite set of n-melements and suppose t ? 0 is an integer. In 1975, P. Erdös asked for the determination of the maximum number of sets in a family $\text{F}$ = {F₁,…, F_m}, F_i ? X, such that ∥F_i ∩ F_j∥ ≠ t for 1 ? i ≠ j ? m. This problem is solved for n ? n₀(t). Let us mention that the case t = 0 is trivial, the answer being 2^{n ? 1}. For t = 1 the problem was solved in [3]. For the proof a result of independent interest (Theorem 1.5) is used, which exhibits connections between linear algebra and extremal set theory.     

Threshold hypergraphs

This paper deals with three generalizations of threshold graphs to hypergraphs proposed by M. Ch. Golumbic. Answering a question of M. Ch. Golumbic we show that these three definitions are not equivalent. The main results of the paper are Theorems 2.5 and 2.6 which characterize hypergraphs satisfying the most general of above definitions.     

On the maximum size of connected hypergraphs without a path of given length

In this note we asymptotically determine the maximum number of hyperedges possible in an $r$-uniform, connected $n$-vertex hypergraph without a Berge path of length $k$, as $n$ and $k$ tend to infinity. We show that, unlike in the graph case, the multiplicative constant is smaller with the assumption of connectivity.     

A short proof of the existence of highly chromatic hypergraphs without short cycles

Using the operation of amalgamation we prove that for every k, n, p there exists a k-graph G (i.e., a k-uniform hypergraph) without cycles of length ?p such that the chromatic number of G is at least n.     

Partitive hypergraphs

We define a new class of hypergraphs (partitive hypergraphs) which generalizes both, the set of all externally related subsets of a graph and the set of all committees of an hypergraph.We give a characterization of the partitive hypergraphs and moreover of those which are associated with hypergraphs or graphs.     

Quasi-random hypergraphs

We introduce an equivalence class of varied properties for hypergraphs. Any hypergraph possessing any one of these properties must of necessity possess them all. Since almost all random hypergraphs share these properties, we term these properties quasi-random. With these results, it becomes quite easy to show that many natural explicit constructions result in hypergraphs which imitate random hypergraphs in a variety of ways.     

Color-bounded hypergraphs, II: Interval hypergraphs and hypertrees

A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set E={E₁,…,E_m}, together with integers s_i and t_i (1≤s_i≤t_i≤|E_i|) for i=1,…,m. A vertex coloring φ is feasible if the number of colors occurring in edge E_i satisfies s_i≤|φ(E_i)|≤t_i, for every i≤m.In this paper we point out that hypertrees-hypergraphs admitting a representation over a (graph) tree where each hyperedge E_i induces a subtree of the underlying tree-play a central role concerning the set of possible numbers of colors that can occur in feasible colorings. We also consider interval hypergraphs and circular hypergraphs, where the underlying graph is a path or a cycle, respectively. Sufficient conditions are given for a ‘gap-free’ chromatic spectrum; i.e., when each number of colors is feasible between minimum and maximum. The algorithmic complexity of colorability is studied, too.Compared with the ‘mixed hypergraphs’-where ‘D-edge’ means (s_i,t_i)=(2,|E_i|), while ‘C-edge’ assumes (s_i,t_i)=(1,|E_i|−1)-the differences are rather significant.     

Series-parallel chromatic hypergraphs

In this paper two-terminal series-parallel chromatic hypergraphs are introduced and for this class of hypergraphs it is shown that the chromatic polynomial can be computed with polynomial complexity. It is also proved that h-uniform multibridge hypergraphs θ(h;a₁,a₂,…,a_k) are chromatically unique for h≥3 if and only if h=3 and a₁=a₂=?=a_k=1, i.e., when they are sunflower hypergraphs having a core of cardinality 2 and all petals being singletons.     

Counting acyclic hypergraphs

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.     

Covers in hypergraphs

Let α(H) denote the stability number of a hypergraphH. The covering number ?(H) is defined as the minimal number of edges fromH to cover its vertex setV(H). The main result is the following extension of König’s wellknown theorem: If α(H′)≧|V(H′)|/2 holds for every section hypergraphH′ ofH then ?(H)≦α(H). This theorem is applied to obtain upper bounds on certain covering numbers of graphs and hypergraphs. In par ticular, we prove a conjecture of B. Bollobás involving the hypergraph Turán numbers.