Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erd o s and R 'enyi, and the family of @math -free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph @math . A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical @math -free graph with @math vertices and @math edges changes as @math grows from @math to @math . In this paper, we resolve this problem in the case when @math is a clique, extending a classical result of Kolaitis, Pr "omel, and Rothschild. In particular, we prove that for every @math , there is an explicit constant @math such that, letting @math , the following holds for every positive constant @math . If @math , then almost all @math -free @math -vertex graphs with @math edges are @math -partite, whereas if @math , then almost all of them are not @math -partite.
Let us remark here that a statement that is even stronger than Conjecture was proved by Osthus, Pr "o mel, and Taraz @cite_3 in the case when @math is a cycle of odd length. More precisely, the following was shown in @cite_3 . Let @math be an integer, let @math be an arbitrary positive constant, and let [ t_ = t_ (n) = ( -1 ( n 2 )^ n )^ 1 -1 . ] If @math , then almost all graphs in @math are not bipartite and if @math , then almost all of them are bipartite.
@cite_3
2128943288 1485502831 2144865572 2070948578
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. 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. We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland's Lower Matching Conjecture and Schrijver's theorem proven by Gurvits and Csikvari. Indeed, our work extends the recent work of Csikvari done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of @math -lifts of the original graph as well as for random @math -lifts of the original graph when @math tends to infinity. We then extend our results to permanents and subpermanents sums. For permanents, we are able to recover the lower bound of Schrijver recently proved by Gurvits using stable polynomials. Our proof is algorithmic and borrows ideas from the theory of local weak convergence of graphs, statistical physics and covers of graphs. We provide new lower bounds for subpermanents sums and obtain new results on the number of matching in random @math -lifts with some implications for the matching measure and the spectral measure of random @math -lifts as well as for the spectral measure of infinite trees. 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.