The number of 4-cycles in a graph

In extremal graph theory studies, it is not yet finished to determine the exact value of ex(n;C4) in general case. Erdős and Simonovits [10] conjectured that an n-vertex graph G with ex(n;C4)+1 edges may contain at least n+o(n) 4-cycles. In this paper we investigate the number of 4-cycles in a graph. We show that the number of 4-cycles in an n-vertex graph G is at least (ϵ4n−3)/4+ϵ2/(2n) provided that |E(G)|=n(1+4n−3)/4+ϵ, where ϵ is an arbitrary positive real number. This result is an approach solving to this open problem. Furthermore, we prove that if n is large enough, then there are at least ((4ϵ−1)4n−3+5)/16+o(1) 4-cycles in an n-vertex graph with e=(n−1)(1+4n−3)/4+ϵ edges which has k vertices whose degrees are mutually different, where k≥24n3+12n−(12n)2−1/273 and ϵ>0 is a real number. In addition, we get that a triangle-free graph with n vertices and e=nn−1/2+ϵ edges has at least (2ϵ(n−1−2)+1)/4+o(1) 4-cycles, where ϵ>0 is an any real number and n is large enough. This shows that the weak version of Erdős and Simonovits' conjecture [10] holds for such graphs when n is sufficiently large. Finally, we obtain a bipartite version of this result.