Results On The Turán Number,Saturation Number,Isolation Number And Leaf Number Of Graphs

This thesis studies several topics in graph theory including the Turán number,sat-uration number,isolation number,leaf number.The main results are as follows.1.Let B_pdenote the book graph consisting of p triangles sharing a common edge.We determine the Turán numbers ex（p+4,B_p）,ex（p+5,B_p）and ex（p+6,B_p）,and characterize the corresponding extremal graphs for the Turán numbers ex（n,B_p）with n=p+2,p+3,p+4,p+5.2.By Simonovits’chromatic critical edge theorem,ex(n,K_m∨C_2k-1)=（?）（（m+1）n²）/（2（m+2））（?） for sufficiently large n.We prove that 2（m+2）k-3（m+2）-1 is large enough for n.3.We determine the saturation number sat（n,P₆+t P₂）for n≥10t/3+10 and characterize the extremal graphs for n>10t/3+20.4.If G is a connected graph with order n≥10,then there exists a subset S（?）V（G）with|S|≤n/5 such that G-N[S]does not contain diamond as a subgraph,where N[S]is the closed neighborhood of S.Furthermore,the bound is sharp.5.Let L（G）denote the leaf number of G.We show that if G is a connected graph of order n with L（G）≤2δ（G）-1,then the circumference of G is at least n-1,and if G is regular then G is hamiltonian.