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

Abstract

Given a graph $T$ and a family of graphs $\mathcal{F}$, the maximum number of copies of $T$ in an $\mathcal{F}$-free graph on $n$ vertices is called the generalized Turán number, denoted by $ex(n, T , \mathcal{F})$. When $T= K_2$, it reduces to the classical Turán number $ex(n, \mathcal{F})$. Let $ex_{bip}(b,n, T , \mathcal{F})$ be the maximum number of copies of $T$ in an $\mathcal{F}$-free bipartite graph with two parts of sizes $b$ and $n$, respectively. Let $P_k$ be the path on $k$ vertices, $\mathcal{C}_{\ge k}$ be the family of all cycles with length at least $k$ and $M_k$ be a matching with $k$ edges. In this article, we determine $ex_{bip}(b,n, K_{s,t}, \mathcal{C}_{\ge 2n-2k})$ exactly in a connected bipartite graph $G$ with minimum degree $δ(G) \geq r\ge 1$, for $b\ge n\ge 2k+2r$ and $k\in \mathbb{Z}$, 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 $ex_{bip}(b,n, K_{s,t}, P_{2n-2k})$ and $ex_{bip}(b,n, K_{s,t}, M_{n-k})$ exactly in a connected bipartite graph $G$ with minimum degree $δ(G) \geq r\ge 1$, which generalizes a theorem of Wang. Moreover, we determine $ex(n, K_{s,t}, \mathcal{C}_{\ge k})$ and $ex(n, K_{s,t}, P_{k})$ respectively in a connected graph $G$ with minimum degree $δ(G) \geq r\ge 1$, which generalizes a theorem of Lu, Yuan and Zhang.