The generalized Tur'{a}n number of long cycles in graphs and bipartite graphs

This is the authors' abstract. We don't add key points for this paper.

Given a graph T and a family of graphs F, the maximum number of copies of T in an F-free graph on n vertices is called the generalized Tur\'{a}n number, denoted by ex(n, T , F). When T= K₂, it reduces to the classical Tur\'{a}n number ex(n, F). Let exbip(b,n, T , F) be the maximum number of copies of T in an F-free bipartite graph with two parts of sizes b and n, respectively. Let Pₖ be the path on k vertices, C≥ k be the family of all cycles with length at least k and Mₖ be a matching with k edges. In this article, we determine exbip(b,n, Ks,t, C≥ 2n-2k) exactly in a connected bipartite graph G with minimum degree δ(G) ≥ r≥ 1, for b≥ n≥ 2k+2r and k∈ Z, which generalizes a theorem of Moon and Moser, a theorem of Jackson and gives an affirmative evidence supporting a conjecture of Adamus and Adamus. As corollaries of our main result, we determine exbip(b,n, Ks,t, P₂ₙ₋₂ₖ) and exbip(b,n, Ks,t, Mₙ₋ₖ) exactly in a connected bipartite graph G with minimum degree δ(G) ≥ r≥ 1, which generalizes a theorem of Wang. Moreover, we determine ex(n, Ks,t, C≥ k) and ex(n, Ks,t, Pₖ) respectively in a connected graph G with minimum degree δ(G) ≥ r≥ 1, which generalizes a theorem of Lu, Yuan and Zhang.