Independent Cycles With Specific Lengths

We discuss independent cycles with specific length and the hamiltonicity in random graphs.Let G be a graph.The order of G is V(G)and its size is E(G).Let v ? V(G).We use d(v,G)to denote the degree of v.The minimum degree of G is denoted by ?(G).The maximum degree of G is denoted by ?(G).We define ?2(G)= min{d(x)+ d(y)\x? V,?V,xy(?)E} If a path(or a cycle)contains all the vertices of G,then define this path(or this cycle)as a Hamilton path(or a hamilton cycle).Independent cycles in a graph is a generalization of hamilton cycle theory.In 1952,Dirac proved that every graph of order n of minimum degree at least[n/2]is hamiltonian.In 1963,Corradi and Hajnal proved the following theorem:let G be a graph of order n with n ? 3k,where k is a positive integer.If the minimum degree ?(G)? 2k,then G contains k independent cycles.In 2004,Wang proved the following result:let G be a graph of order n with 4k +1 ? n ? 4k + 4,where k is a positive integer.If the minimum degree ?(G)? 2k:+ 1,then G contains k independent quadrilaterals.In this paper,we prove the following results:Result 1:Let G be a simple graph of order n with 4k + 1 ? n ? 4k+ 1+where k is a positive integer.If ?2(G)? n,then G contains k independent quadrilaterals.Result 2:Let G be a graph of order n>4k,where k is a positive integer.If?(G)? + 1,then G has a 2-factor with k cycles,such that k-1 of them are quadrilaterals.Many experts and scholars focus on random graph.Let G = G(n,p)denote the random graph with n vertices and edge probability p.Lee and Sudakov proved that if p>>lnn/n,then a.a.s every subgraph of G(n,p)with minimum degree at least(1/2 + o(1))np is hamiltonian.In 2016 Shang proved this result:for any ?>0.and if p>>ln n/n,then a.a.s every subgraph of G(n,n,p)with minimum degree at least(1/2 + o(1))np is hamiltonian.We prove the following theorem:Result 3:For ?>0,if there exists C = C(?)such that p ? Clnn/n,then every subgraph of G(n,...,n,p)with minimum degree at least ?(G)is hamiltonian,where ?(G)is as follows:...