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.