Some Extremal Problems On Cycle Length

Let n,r,t be positive integers and let G be a simple,connect and undirected graph of order n.A graph G of order n is said to be r-(p₀,…,p_t-1)-pancyclic if G contains exactly pi（0≤i≤t-1）cycles Cr+tj+i for r+tj+t-1≤n and j is the number of p₀,…,p_t-1 repeats.A graph G of order n is said to be r-(p₀,…,p_t-1)-oddpancyclic（or bipancyclic） if G contains exactly pi（0≤i≤f-1） cycles Cr+tj+i for r+tj+t-1≤n and j is the number of P0,…,p_t-1 repeats.The cycle length distribution of a graph of order n is（c₁,c₂,…,cn）,where ci is the number of cycles of length i,for i=1,2,…,n.Let g（a1,…,an）denote the minimum possible number of edges in a graph which satisfies c_i≥a_i where ai is a nonnegative integer.In this paper,we mainly consider some extremal problems on the cycle length of graphs.The following are our main results:1.We provide a construction for r-（p₀,…,p₇）- pancyclic graphs.Similar method are used to construct r-（p₀,…,p₇）- oddpancyclic （or bipancyclic） graphs. Moreover,we also construct a class of graphs with cycle length distribution （0,0,c₃,c₄,…,c_n）,where ci≥ai,i=3,4,…,n and give the lower bounds for g（0,0,a₃,a₄,…,a_n）,respectively；2.We determine the minimum possible size of simple graphs of order n with at least two t-cycles for each t∈{3,…,n},where 3≤n≤19.