Research On Some Turán-type Problems And Related Stability Results

Turan problem is one of the most important research topic in extremal combinatorics.The main problem is to determine the maximum number of edges in a graph which does not contain a fixed forbidden subgraph.If we determine the maximum number of edges of a graph without a fixed forbidden subgraph,then further study the structure of a graph without the given forbidden subgraph when its number of edges is close to the maximum number of edges,which is usually called the stability problem.In the past decades,the related stability of many extremal problems has been widely studied.In this paper,we mainly study the stability of odd cycles and prove two new results.In addition,we also consider the generalized Turan problem.The specific contents are as follows:In Chapter 2,we study the stability of problem on maximal odd cycles.For any positive integer k,we show that every maximal C_2k+1-free graph with at least n²/4-o(n^3/2)edges contains an induced complete bipartite subgraph on（1-o（1））n vertices.We also show that this is best possible.In Chapter 3,we study the stability of problem on odd cycles.We show that if G is a C_2k-1-free graph on n vertices with at least n²/4-δn² edges,then 4/4k²-12k+9δ²n²≤g_k（n,o（n²））≤ 260δ²n².In Chapter 4,we study the generalized Turán number on ex（n,K_r,2K_r）.We show that ex（n,K_r,2K_r）=N(K_r-1,T_r-1（n-1）)for n≥ 3r5 and r≥ 3,where N(K_r-1,T_r-1（n-1）)denotes the number of（r-1）-cliques in（r-1）-partite Turán graph on n-1 vertices.For r=3,we determine ex（n,K₃,2K₃）for all n.