Study On Several Extremal Problems Under Degree Conditions

To determine the bound of some parameters of hypergraphs with given conditions is an interesting problem.On the basis of previous work,this thesis focus on several specific extremal type problems.The major content of this thesis can be concluded as follows:In section 1,we introduce some basic definitions and primary results,and expound the background of our research.In section 2,we discuss a specific Turán type problem.Chvatal and Hanson gave a tight upper bound of the size of graphs with restricted maximum degree and matching number;Khare studied the same problem for linear 3-graphs with restricted matching number and maximum degree.In this paper,we give a upper bound of the size of 3-graphs with bounded codegree and matching number,and the upper is tight according to the constructed extremal 3-graphs.We then consider a tiling problem in section 3.Given integer k and a k-graph F,let t_k-1?n,F?be the minimum integer t such that every k-graph H on n vertices with codegree at least t contains an F-factor.For integers k? 3 and 0 ?1,let y_k,l be a k-graph with two edges that shares exactly l vertices.Han and Zhao asked the following question:For all k?3,0?l?k-1 and sufficiently large n divisible by 2k-l,determine the exact value of t_k-1(n,y_k,1).In this paper,we determine the exact value of t_k-1(n,y_k,1)for k?3 and 1?l?k-2,combining with two previously known results of Rodl,Rucinski and Szemeredi and Gao,Han and Zhao,the question of Han and Zhao is solved completely.In section 4,we construct two extremal 3-graphs for covering problems.Given two 3-graphs F and H,an F-covering of H is a collection of copies of F in H such that each vertex of H is contained in at least one copy of them.Let c₂?n,F?be the maximum integer t such that every 3-graph with minimum codegree greater than t has an F-covering.In this paper,we determine the exact value of c₂(n,K_t^3-)and c₂(n,K₅^3-),where K_t^3-is the complete 3-graph on t vertices with one edge removed.Through these results,an open problem proposed by Falgas-Ravry and Zhao is solved completely.A long-standing conjecture asserts that there exists a constant c>0 such that every graph of order n without isolated vertices contains an induced subgraph of order at least cn with all degrees odd.Scott proved that every graph G has an induced subgraph of order at least |V?G?|/?2??G??with all degrees odd,where ??G?is the chromatic number of G,this implies the conjecture for graphs with bounded chromatic number.But the factor1/?2X?G??seems to be not best possible,for example,Radcliffe and Scott proved c=2/3 for trees,Berman,Wang and Wargo showed that c=2/5 for graphs with maximum degree 3,so it is interesting to determine the exact value of c for special family of graphs.In this paper,we further confirm the conjecture for graphs with treewidth at most 2 with c=2/5,and the bound is best possible.