Let @math be a connected graph of order @math , whose minimum vertex degree is at least @math . A subset @math of vertices in @math is a @math -tuple total dominating set if every vertex of @math is adjacent to at least @math vertices in @math . The minimum cardinality of a @math -tuple total dominating set of @math is the @math -tuple total domination number of @math , denoted by @math . Henning and Yeo in hen proved that if @math is a cubic graph different from the Heawood graph, @math , and this bound is sharp. Similarly, a @math -tuple dominating set is a subset @math of vertices of @math , @math such that @math for every vertex @math , where @math . The @math -tuple domination number of @math , denoted by @math , is the minimum cardinality of a @math -tuple dominating set of @math . In this paper, we give a simple approach to compute an upper bound for @math -tuple total domination number of @math -regular graphs. Also, we give an upper bound for the @math -tuple dominating number of @math -regular graphs. In addition, our method gives algorithms to compute dominating sets with the given bounds, while the previous methods are existential.
Let @math be the Heawood graph (or, equivalently, the incidence bipartite graph of the Fano plane) on @math vertices shown in Figure . In @cite_7 , Henning and Yeo proved some theorems about strong transversal in hypergraphs and then as an application of their hypergraph results they proved the following theorem.
@cite_7
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Let @math be a 3-uniform hypergraph of order @math and size @math , and let @math be a subset of vertices of @math . The set @math is a strong transversal in @math if @math contains at least two vertices from every edge of @math . The strong transversal number @math of @math is the minimum size of a strong transversal in @math . We show that @math , and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph @math of order @math and size @math that achieves equality in this bound. A set @math of vertices in a graph @math is a double total dominating set of @math if every vertex of @math is adjacent to at least two vertices in @math . The minimum cardinality of a double total dominating set of @math is the double total domination number @math of @math . Let @math be a connected graph of order @math with minimum degree at least three. As an application of our hypergraph results, we show that @math with equality if and only if @math is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if @math is not the Heawood graph, we show that @math , while if @math is a cubic graph different from the Heawood graph, we show that @math , and this bound is sharp. Friedland's Lower Matching Conjecture asserts that if @math is a @math --regular bipartite graph on @math vertices, and @math denotes the number of matchings of size @math , then @math where @math . When @math , this conjecture reduces to a theorem of Schrijver which says that a @math --regular bipartite graph on @math vertices has at least @math perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that @math In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra @math factor compared to the conjecture if @math is separated away from @math and @math , and is tight up to a constant factor if @math is separated away from @math . We will also give a new proof of Gurvits's and Schrijver's theorems, and we extend these theorems to @math --biregular bipartite graphs. Let H d (n,p) signify a random d -uniform hypergraph with n vertices in which each of the @math possible edges is present with probability p = p(n) independently, and let H d (n,m) denote a uniformly distributed d -uniform hypergraph with n vertices and m edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of H d (n,p) and H d (n,m) in the regime @math , resp. d(d−1)m n >1+ϵ, where ϵ>0 is arbitrarily small but fixed as n → ∞. The proofs are based on a purely probabilistic approach. Denote by an @math -component a connected @math -uniform hypergraph with @math edges and @math vertices. We prove that the expected number of creations of @math -component during a random hypergraph process tends to 1 as @math and @math tend to @math with the total number of vertices @math such that @math . Under the same conditions, we also show that the expected number of vertices that ever belong to an @math -component is approximately @math . As an immediate consequence, it follows that with high probability the largest @math -component during the process is of size @math . Our results give insight about the size of giant components inside the phase transition of random hypergraphs. We prove an incidence theorem for points and planes in the projective space @math over any field @math , whose characteristic @math An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving @math if @math . This yields a bound on the number of incidences between @math points and @math planes in @math , with @math as @math where @math is the maximum number of collinear planes, provided that @math if @math . Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when @math . For a non-collinear point set @math and a non-degenerate symmetric or skew-symmetric bilinear form @math , the number of distinct values of @math on pairs of points of @math is @math . This is also the best known bound over @math , where it follows from the Szemer 'edi-Trotter theorem. Also, a set @math , not supported in a single semi-isotropic plane contains a point, from which @math distinct distances to other points of @math are attained. A theorem of A. Schrijver asserts that a d-regular bipartite graph on 2n vertices has at least @math perfect matchings. L. Gurvits gave an extension of Schrijver’s theorem for matchings of density p. In this paper we give a stronger version of Gurvits’s theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive integer k, there exists a positive constant c(k) such that if a d-regular vertex-transitive bipartite graph on 2n vertices contains a cycle of length at most k, then it has at least @math perfect matchings. Given a t-uniform hypergraph H on k vertices and an assignment of integers f(T) to thet -subsets T of a v-setX , v k+t, e give necessary and sufficient conditions for the existence of an assignment of integer multiplicities h(G) to those subhypergraphs G of the complete t-uniform hypergraph on v vertices that are isomorphic to H so that the sum of the integers h(G) over those G that contain T is f(T). Our main theorem is stated in terms of integral matrices. As a consequence of our main theorem, e also determine a diagonal form, and hence the p-rank for all primes p, for the incidence matrix of t-subsets versus subhypergraphs isomorphic to H. We examine several online matching problems, with applications to Internet advertising reservation systems. Consider an edge-weighted bipartite graph G, with partite sets L, R. We develop an 8-competitive algorithm for the following secretary problem: Initially given R, and the size of L, the algorithm receives the vertices of L sequentially, in a random order. When a vertex l L is seen, all edges incident to l are revealed, together with their weights. The algorithm must immediately either match l to an available vertex of R, or decide that l will remain unmatched. Dimitrov and Plaxton show a 16-competitive algorithm for the transversal matroid secretary problem, which is the special case with weights on vertices, not edges. (Equivalently, one may assume that for each l L, the weights on all edges incident to l are identical.) We use a similar algorithm, but simplify and improve the analysis to obtain a better competitive ratio for the more general problem. Perhaps of more interest is the fact that our analysis is easily extended to obtain competitive algorithms for similar problems, such as to find disjoint sets of edges in hypergraphs where edges arrive online. We also introduce secretary problems with adversarially chosen groups. Finally, we give a 2e-competitive algorithm for the secretary problem on graphic matroids, where, with edges appearing online, the goal is to find a maximum-weight acyclic subgraph of a given graph. Abstract An elegant and general way to apply graph partitioning algorithms to hypergraphs would be to model hypergraphs by graphs and apply the graph algorithms to these models. Of course such models have to simulate the given hypergraphs with respect to their cut properties. An edge-weighted graph ( V , E ) is a cut-model for an edge-weighted hypergraph ( V , H ) if the weight of the edges cut by any bipartition of V in the graph is the same as the weight of the hyperedges cut by the same bipartition in the hypergraph. We show that there is no cut-model in general. Next we examine whether the addition of dummy vertices helps: An edge-weighted graph ( V ∪ D , E ) is a mincut-model for an edge-weighted hypergraph ( V, H ) if the weight of the hyperedges cut by a bipartition of the hypergraphs vertices is the same as the weight of a minimum cut separating the two parts in the graph. We construct such models using positive and negative weights. On the other hand, we show that there is no mincut-model in general if only positive weights are allowed. Computing a minimum vertex cover in graphs and hypergraphs is a well-studied optimizaton problem. While intractable in general, it is well known that on bipartite graphs, vertex cover is polynomial time solvable. In this work, we study the natural extension of bipartite vertex cover to hypergraphs, namely finding a small vertex cover in k- uniform k-partite hypergraphs, when the k-partition is given as input. For this problem Lovasz [16] gave a k 2 factor LP rounding based approximation, and a matching (k 2 - o(1)) integrality gap instance was constructed by [1]. We prove the following results, which are the first strong hardness results for this problem (heree > 0 is an arbitrary constant): - NP-hardness of approximating within a factor of (k 4 -e), and - Unique Games-hardness of approximating within a factor of (k 2 -e), showing optimality of Lovasz's algorithm under the Unique Games conjecture. The NP-hardness result is based on a reduction from minimum vertex cover in r-uniform hypergraphs for which NP-hardness of approximating within r-1-e was shown by [5]. The Unique Games-hardness result is obtained by applying the recent results of [15], with a slight modification, to the LP integrality gap due to [1]. The modification is to ensure that the reduction preserves the desired structural properties of the hypergraph. We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most @math , the algorithm achieves a performance ratio of @math for large @math , which improves the previously known ratio of @math obtained by Halld orsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For k-uniform hypergraphs with n vertices, we achieve a ratio of @math for large n, and for k-uniform hypergraphs with maximum degree at most @math the algorithm achieves a ratio of @math for large @math . These results considerably improve the previous best ratio of @math for bounded degree k-uniform hypergraphs, and @math for general k-uniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldorsson. It is NP-Hard to find a proper 2-coloring of a given 2-colorable (bipartite) hypergraph H. We consider algorithms that will color such a hypergraph using few colors in polynomial time. The results of the paper can be summarized as follows: Let n denote the number of vertices of H and m the number of edges, (i) For bipartite hypergraphs of dimension k there is a polynomial time algorithm which produces a proper coloring using min ( O(n^ 1 - 1 k ),O((m n)^ 1 k - 1 ) )colors, (ii) For 3-uniform bipartite hypergraphs, the bound is reduced to O(n2 9). (iii) For a class of dense 3-uniform bipartite hypergraphs, we have a randomized algorithm which can color optimally. (iv) For a model of random bipartite hypergraphs with edge probability p≥ dn−2, d > O a sufficiently large constant, we can almost surely find a proper 2-coloring. Suppose we are given a bipartite graph with vertex set X, Y, |X| = n , |Y| = N , each point in X (Y) has degree D(d) fixed, respectively, moreover, each pair of points x, x' ∈ X has at most D (log n) 3 (common) neighbours. Let t (X, Y) denote the minimum number of vertices of Y needed to cover all vertices of X. We prove (Theorem 1.1) that t (X, Y) d n tends to 1 as n tends to infinity. This result has many applications: Theorem [5]. Suppose k > r > 1 are fixed, n → ∞. Then there exists a collection of (1 + o (1)) × ( n r ) ( k r ) k -subsets of an n -set so that each r -subset is contained in at least one member of the collection. Analogues and strengthenings of this result are deduced. E.g. for vector spaces, orthogonal or simplectic geometries, random collections of k -sets with constant probabilities, etc. Theorem 3.3. Suppose G is a graph on v vertices and e edges and ℛ is a random graph on n vertices and edge probability e ( v 2 ). Then there exists a collection of (1 + o (1))( n 2 ) ( v 2 ) induced subgraphs of ℛ on v vertices, isomorphic to G and such that each edge (non-edge) of ℛ is covered by an edge (non-edge) of a graph in the collection. We consider the problem of @math -colouring a random @math -uniform hypergraph with @math vertices and @math edges, where @math , @math , @math remain constant as @math tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case @math , must have one of two easily computable values as @math tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs. Denote by @math the class of all triangle-free graphs on n vertices and m edges. Our main result is the following sharp threshold, which answers the question for which densities a typical triangle-free graph is bipartite. Fix e > 0 and let @math . If n 2 ≤ m ≤ (1 − e) t3, then almost all graphs in @math are not bipartite, whereas if m ≥ (1 + e)t3, then almost all of them are bipartite. For m ≥ (1 + e)t3, this allows us to determine asymptotically the number of graphs in @math . We also obtain corresponding results for Cl-free graphs, for any cycle Cl of fixed odd length. For the set of graphs with a given degree sequence, consisting of any number of @math and @math , and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of @math -matchings. We find the expected value of the number of @math -matchings of @math -regular bipartite graphs on @math vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for @math -matchings in @math -regular bipartite graphs on @math vertices, and their asymptotic versions for infinite @math -regular bipartite graphs. We prove these conjectures for @math -regular bipartite graphs and for @math -matchings with @math . Abstract We would like to classify the vertices of a hypergraph in the way that ‘similar’ vertices (those having many incident edges in common) belong to the same cluster. The problem is formulated as follows: given a connected hypergraph on n vertices and fixing the integer k (1 k ⩽ n ), we are looking for k -partition of the set of vertices such that the edges of the corresponding cut-set be as few as possible. We introduce some combinatorial measures characterizing this structural property and give upper and lower bounds for them by means of the k smallest eigenvalues of the hypergraph. For this purpose the notion of spectra of hypergraphs — which is the generalization of C -spectra of graphs — is also introduced together with k-dimensional Euclidean representations . We shall that the existence of k 'small' eigenvalues is a necessary but not sufficient condition for the existence of a good clustering. In addition the representatives of the vertices in an optimal k -dimensional Euclidean representation of the hypergraph should be well separated by means of their Euclidean distances. In this case the k -partition giving the optimal clustering is also obtained by this classification method. We establish central and local limit theorems for the number of vertices in the largest component of a random d-uniform hypergraph Hd(n,p) with edge probability p = c- @math , where c > (d - 1)-1 is a constant. The proof relies on a new, purely probabilistic approach. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010 A hypergraph H = (V,E) is called s-orientable, if there is an assignment of each edge e ∈ E to one of its vertices v ∈ e such that no vertex is assigned more than s edges. Let Hn,m,k be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the 1-orientability of Hn,m,k for all k ≥ 3, i.e., we determine a critical quantity ck* such that with probability 1 - o(1) the graph Hn,cn,k has a 1-orientation if c ck*. We present two applications of this result that involve the paradigm of multiple choices. First, we show how it implies sharp load thresholds for cuckoo hash tables, where each element chooses k out of n locations. Particularly, for each k ≥ 3 we prove that with probability 1 - o(1) the maximum number of elements that can be hashed is (1 - o(1))ck*n, and more items prevent the successful allocation. Second, we study random graph processes, where in each step we have the choice among any edge connecting k random vertices. Here we show the existence of a phase transition for avoiding a giant connected component, and quantify precisely the dependence on k. Let h>w>0 be two fixed integers. Let H be a random hypergraph whose hyperedges are all of cardinality h. To w-orient a hyperedge, we assign exactly w of its vertices positive signs with respect to the hyperedge, and the rest negative. A (w,k)-orientation of H consists of a w-orientation of all hyperedges of H, such that each vertex receives at most k positive signs from its incident hyperedges. When k is large enough, we determine the threshold of the existence of a (w,k)-orientation of a random hypergraph. The (w,k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when h=2 and w=1, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran, thereby settling a conjecture made by Karp and Saks. Motivated by a problem of cuckoo hashing, the special hypergraph case with w=k=1, was solved in three separate preprints dating from October 2009, by Frieze and Melsted, by Fountoulakis and Panagiotou, and by Dietzfelbinger, Goerdt, Mitzenmacher, Montanari, Pagh and Rink.