The generalized Tur'an number of long cycles in graphs and bipartite graphs

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\'an number, denoted by ex (n, T, F). When T= K₂, it reduces to the classical Tur\'an number ex (n, F). Let ex₁₈₏ (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 ₊ be the family of all cycles with length at least k and Mₖ be a matching with k edges. In this article, we determine ex₁₈₏ (b, n, Kₒ, ₓ, C ₂₍-₂₊) exactly in a connected bipartite graph G with minimum degree (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 ex₁₈₏ (b, n, Kₒ, ₓ, P₂₍-₂₊) and ex₁₈₏ (b, n, Kₒ, ₓ, M₍-₊) exactly in a connected bipartite graph G with minimum degree (G) r 1, which generalizes a theorem of Wang. Moreover, we determine ex (n, Kₒ, ₓ, C ₊) and ex (n, Kₒ, ₓ, P₊) respectively in a connected graph G with minimum degree (G) r 1, which generalizes a theorem of Lu, Yuan and Zhang.