The Turán Number Of P₅?P_2l+1

Extremal graph theory is a branch of combinatorics mathematics.It mainly studies the extremal values of some parameters for a given family of graph.The Turan number mainly discussed in this paper belongs to the extremal problem in graph theory.The Turan number of a graph H,denoted by ex(n,H),is the maximum number of edges in any graph of order n which does not contain H as a subgraph.Yuan and Zhang proposed a conjecture on ex(n,Fm)in 2016,where Fm is the disjoint union of m paths.Motivated by the conjecture,we determine ex(n,P5 ? P21+1)in connected graphs and disconnected graphs of order n when G does not contian P2l+5.The organization of this theis is following:Firstly,we briefly introduce the research background and current status of the Turan number and give the definitions of this paper.Moreover,we consider the Turan number of P5 ? P2l+1 in graphs of order n.When G does not contian P21+5,we determine ex(n,P5 ? P2l+1)and the extremal graphs for n? 2l+6,which partially confirms the conjecture.For l=3 and l=4,we consider the Turan number of P5 ? P7 and P5 ? P9 in connected graphs and disconnected graphs of order n.We determine the exact value of ex(n,P5 ? P7),ex(n,P5 ? P9)and the extremal graphs.Lastly,we summarize the main work of this paper and discuss future work.