一致超图中Berge线性森林的Turán数

Turán Number of Berge Linear Forests in Uniform Hypergraphs

设F是一个图,H是一个超图,如果存在一个双射Φ:E(F)→E(H),使得■e∈E(F)有e■Φ(e),那么称超图H是Berge-F.不含Berge-F作为子超图的n阶r-一致超图所能达到的最大边数称为Berge-F的Turán数,记作ex_(r)(n,Berge-F).线性森林是指连通分支全是路或者孤立顶点的图.设L_(n,k)是一类含有n个顶点k条边的线性森林图族.本文研究了r-一致超图中Berge-L_(n,k)的Turán数.当r≥k+1和3≤r≤[(k-1)/2]-1时,分别确定了ex_(r)(n,Berge-L_(n,k))的精确值;当[(k-1)/2]≤r≤k时,给出了ex_(r)(n,Berge-L_(n,k))的上界.

Let F be a graph and H be a hypergraph.H is called a Berge-F if there is a bijectionΦ:E(F)→E(H)such that for each e∈E(F)we have e■Φ(e).The maximum possible number of hyperedges in an n-vertex r-uniform hypergraph containing no Berge-F as its subhypergraph is called the Turán number of Berge-F,denoted by ex_(r)(n,Berge-F).A linear forest is a graph whose connected components are paths or isolated vertices.Let L_(n,k)be the family of all linear forests on n vertices with k edges.In this paper we determine the Turán number of Berge-L_(n,k)in r-uniform hypergraphs.When r≥k+1 or 3≤r≤[(k-1)/2]-1,we determine the exact value of ex_(r)(n,Berge-L_(n,k))respectively.When[(k-1)/2]≤r≤k,we determine the upper bound of ex_(r)(n,Berge-L_(n,k)).