General Hypergraphs Research Articles

Equicovering Subgraphs of Graphs and Hypergraphs

As a variation on the $t$-Equal Union Property ($t$-EUP) introduced by Lindström, we introduce the $t$-Equal Valence Property ($t$-EVP) for hypergraphs: a hypergraph satisfies the $t$-EVP if there are $t$ pairwise edge-disjoint subhypergraphs such that for each vertex $v$, the degree of $v$ in all $t$ subhypergraphs is the same. In the $t$-EUP, the subhypergraphs just have the same sets of vertices with positive degree. For both the $2$-EUP and the $2$-EVP, we characterize the graphs satisfying the property and determine the maximum number of edges in a graph not satisfying it. We also study the maximum number of edges in both $k$-uniform and general hypergraphs not satisfying the $t$-EVP.

The colorful Helly theorem and general hypergraphs

The definition of the Helly property for hypergraphs was motivated by the Helly theorem for convex sets. Similarly, we define the colorful Helly property for a family of hypergraphs, motivated by the colorful Helly theorem for collections of convex sets, by Lovász. We describe some general facts about the colorful Helly property and prove complexity results. In particular, we show that it is Co-NP-complete to decide if a family of p hypergraphs is colorful Helly, even if p=2. However, for any fixed p, we describe a polynomial time algorithm to decide if such family is colorful Helly, provided at least p−1 of the hypergraphs are p-Helly.

Cellular resolutions of cointerval ideals

Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d = 2), and strictly contains the class of ‘strongly stable’ hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs.

Conflict-Free Colourings of Graphs and Hypergraphs

A colouring of the vertices of a hypergraphHis calledconflict-freeif each hyperedgeEofHcontains a vertex of ‘unique’ colour that does not get repeated inE. The smallest number of colours required for such a colouring is called theconflict-free chromatic numberofH, and is denoted by χCF(H). This parameter was first introduced by Even, Lotker, Ron and Smorodinsky (FOCS2002) in ageometricsetting, in connection with frequency assignment problems for cellular networks. Here we analyse this notion for general hypergraphs. It is shown that$\chi_{\rm CF}(H)\leq 1/2+\sqrt{2m+1/4}$, for every hypergraph withmedges, and that this bound is tight. Better bounds of the order ofm1/tlogmare proved under the assumption that the size of every edge ofHis at least 2t− 1, for somet≥ 3. Using Lovász's Local Lemma, the same result holds for hypergraphs in which the size of every edge is at least 2t− 1 and every edge intersects at mostmothers. We give efficient polynomial-time algorithms to obtain such colourings.Our machinery can also be applied to the hypergraphs induced by the neighbourhoods of the vertices of a graph. It turns out that in this case we need far fewer colours. For example, it is shown that the vertices of any graphGwith maximum degree Δ can be coloured with log2+εΔ colours, so that the neighbourhood of every vertex contains a point of ‘unique’ colour. We give an efficient deterministic algorithm to find such a colouring, based on a randomized algorithmic version of the Lovász Local Lemma, suggested by Beck, Molloy and Reed. To achieve this, we need to (1) correct a small error in the Molloy–Reed approach, (2) restate and re-prove their result in adeterministicform.

Self-complementary hypergraphs and their self-complementing permutations

Abstract 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 ′ = ( V ; ( V k ) − 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, given independently in the years 1962 – 1963 by Sachs and Ringel, and those obtained for 3–uniform hypergraphs by Kocay, for 4–uniform hypergraphs by Szymanski, and for general (not uniform) hypergraphs by Zwonek.

Multicolour Discrepancies

In this article we introduce combinatorial multicolour discrepancies and generalize several classical results from $2$-colour discrepancy theory to $c$ colours ($c \geq 2$). We give a recursive method that constructs $c$-colourings from approximations of $2$-colour discrepancies. This method works for a large class of theorems, such as the ‘six standard deviations’ theorem of Spencer (1985), the Beck–Fiala (1981) theorem, the results of Matoušek, Wernisch and Welzl (1994) and Matoušek (1995) for bounded VC-dimension, and Matoušek and Spencer's (1996) upper bound for the arithmetic progressions. In particular, the $c$-colour discrepancy of an arbitrary hypergraph ($n$ vertices, $m$ hyperedges) is \[ \OO\Bigl(\sqrt{\tfrac n c\,\log m}\Bigr). \] If $m = \OO(n)$, then this bound improves to \[ \OO\Bigl(\sqrt{\tfrac n c\,\log c}\Bigr). \]On the other hand there are examples showing that discrepancy in $c$ colours can not be bounded in terms of two-colour discrepancies in general, even if $c$ is a power of 2. For the linear discrepancy version of the Beck–Fiala theorem, the recursive approach also fails.Here we extend the method of floating colours via tensor products of matrices to multicolourings, and prove multicolour versions of the Beck–Fiala theorem and the Bárány–Grinberg theorem. Using properties of the tensor product we derive a lower bound for the $c$-colour discrepancy of general hypergraphs. For the hypergraph of arithmetic progressions in $\{1, \ldots, n\}$ this yields a lower bound of $\frac{1}{25 \sqrt c} \sqrt[4]{n}$ for the discrepancy in $c$ colours. The recursive method shows an upper bound of $\OO(c^{-0.16} \sqrt[4]{n})$

New Lower Bounds for Ramsey Numbers of Graphs and Hypergraphs

Let G be an r-uniform hypergraph. The multicolor Ramsey number rk(G) is the minimum n such that every k-coloring of the edges of the complete r-uniform hypergraph K(r)n yields a monochromatic copy of G. Improving slightly upon results from M. Axenovich, Z. Füredi, and D. Mubayi (J. Combin. Theory Ser. B79 (2000), 66–86), we prove thattk2+1≤rk(K2,t+1)≤tk2+k+2,where the lower bound holds when t and k are both powers of a prime p. When t=1, we improve the lower bound by 1, proving that rk(C4)≥k2+2 for any prime power k. This extends the result of F. Lazebnik and A. J. Woldar (J. Combin. Theory Ser. B79 (2000), 172–176), which proves the same bound when k is an odd prime power. These results are generalized to hypergraphs in the following sense. Fix integers r,s,t≥2. Let H(r)(s,t) be the complete r-partite r-graph with r−2 parts of size 1, one part of size s, and one part of size t (note that H(2)(s,t)=Ks,t). We prove thattk2−k+1≤rk(H(r)(2,t+1))≤tk2+k+r,where the lower bound holds when t and k are both powers of a prime p and(1−o(1))ks≤rk(H(r)(s,t))≤O(ks),for fixedt,s≥2,t>(s−1)!,rk(H(r)(3,3))=(1+o(1))k3.Some of our lower bounds are special cases of a family of more general hypergraph constructions obtained by algebraic methods. We describe these, thereby extending results of F. Lazebnik and A. J. Woldar (J. Graph Theory, 2001) about graphs.

An Approximation Algorithm for Hypergraph Max k-Cut with Given Sizes of Parts

<p>Probably most of the recent striking breakthroughs in designing approximation algorithms with provable performance guarantees are due to using novel methods of rounding polynomially solvable fractional relaxations. Applicability of the known rounding methods is highly dependent on the type of the constraints in such relaxations. In [1] the authors presented a new rounding ( pipage) method especially oriented to tackle some NP-hard problems which can be equivalently reformulated as integer programs with cardinality or a bit more general constraints. The paper [1] contains four results demonstrating<br />the strength of the pipage rounding. One of them is an 1/2-approximation algorithm for Max k-Cut with given sizes of parts. An instance of this problem consists of an undirected graph G = (V,E), a collection of nonnegative weights w_e associated with its edges and k positive integers p1, p2, . . . , pk such that Sum pi = |V|. It is required to find a partition of V into k parts V1, V2, . . . , Vk with each part Vi having size pi so as to maximize the total weight of edges whose ends lie in different parts of the partition. The Max<br />Cut and Max k-Cut problems are classical in combinatorial optimization and<br />have been extensively studied in the absence of cardinality constraints. The<br />best known approximation algorithm for Max Cut is due to Goemans and<br />Williamson [8] and has performance guarantee of 0.878. Frieze and Jerrum<br />[7] extended the technique of Goemans and Williamson to Max k-Cut and<br />designed a (1−1/k+2 ln k/k^2)-approximation algorithm. Few approximation<br />algorithms are known for some special cases of Max k-Cut with given sizes<br />of parts. In particular, Frieze and Jerrum [7] present an 0.65-approximation<br />algorithm for Max Bisection (in this problem k = 2 and p1 = p2 = |V|/2).<br />Very recently, Ye [9] announced an algorithm with a better performance guarantee<br />of 0.699. The best known approximation algorithm for Max k-Section<br />(in this problem p1 = ... = pk = |V|/k) is due to Andersson [2] and has<br />performance guarantee of 1 − 1/k + Theta(1/k^3). In this paper we consider a<br />natural hypergraph generalization of Max k-Cut with given sizes of parts<br />| - Hypergraph Max k-Cut with given sizes of parts (HMkC for short). An<br />instance of HMkC consists of a hypergraph H = (V,E), a collection of nonnegative<br />weights wS on its edges S, and k positive integers p1, . . . , pk such<br />that Sum pi = |V|. It is required to partition the vertex set V into k parts<br />(X1, . . . , Xk) with |Xi| = pi for each i, so as to maximize the total weight<br />of edges of H not lying wholly in any part of the partition (that is, to maximize<br />the total weight of edges S such that S \ Xi 6 |= 0 for each i). Several<br />closely related versions of Hypergraph Max k-Cut were studied in the literature<br />but very few results have been obtained. Andersson and Engebretsen<br />[3] presented an 0.72-approximation algorithm for the ordinary Hypergraph<br />Max Cut problem. Arora, Karger and Karpinski [4] designed a PTAS for<br />dense instances of this problem (i.e. in the case of hypergraphs H having<br />Theta(|V (H)|^d) edges) under the condition that |S| <= d for each edge S and<br />some constant d.<br />In this paper by applying the pipage rounding method we prove that<br />HMkC can be approximated within a factor of minfjSj : S 2 Eg of the<br />optimum where r = 1−(1−1=r)r−(1=r)r. By direct calculations it easy to<br />get some specic values of r: 2 = 1=2, 3 = 2=3 0:666, 4 = 87=128 <br />0:679, 5 = 84=125 = 0:672, 6 0:665 and so on. It is clear that r tendsto 1 − e−1 0:632 as r ! 1. A less trivial fact is that r > 1 − e−1 for each r 3 (Lemma 2 in this paper). Adding up we arrive at the following conclusions: our algorithm nds a feasible cut of weight within a factor of 1=2 on general hypergraphs (we assume that each edge in a hypergraph has size at least 2), and within a factor of 1 − e−1 in the case when each edge has size at least 3. Note that the rst bound coincides with that we obtained in [1] for the case of graphs. In this paper we also show that in the case of hypergraphs without two-vertex edges the bound of 1 − e−1 cannot be improved unless P=NP.</p>

Asymptotics of the Chromatic Index for Multigraphs

For a multigraph G , let D ( G ) denote maximum degree and set[formula]We show that the chromatic index χ ′( G ) is asymptotically max{ D ( G ), Ggr; ( G )}. The latter is, by a theorem of Edmonds (1965), the fractional chromatic index of G , and the asymptotics established here are part of a conjecture of the author predicting similar agreement of fractional and ordinary chromatic indices for more general hypergraphs. Of particular interest in the present proof is the use of probabilistic ideas and “hard-core” distributions to go from fractional to ordinary (integer) colorings.

Descriptions of state spaces of orthomodular lattices (the hypergraph approach)

Using the general hypergraph technique developed in [7], we first give a much simpler proof of Shultz's theorem [10]: Each compact convex set is affinely homeomorphic to the state space of an orthomodular lattice. We also present partial solutions to open questions formulated in [10] - we show that not every compact convex set has to be a state space of a unital orthomodular lattice and that for unital orthomodular lattices the state space characterization can be obtained in the context of unital hypergraphs.

Fractional matchings and covers in infinite hypergraphs

A strong version of the duality theorem of linear programming is proved for fractional covers and matchings in countable graphs. It is conjectured to hold for general hypergraphs. In Section 2 we show that in countable hypergraphs there does not necessarily exist a maximal matchable set, contrary to the situation in graphs.