The Generalized Turán Number of Spanning Linear Forests

Abstract

Let ${\mathcal{F}}$ be a family of graphs. A graph G is called ${\mathcal{F}}$-free if for any $F\in {\mathcal{F}}$, there is no subgraph of G isomorphic to F. Given a graph T and a family of graphs ${\mathcal{F}}$, the generalized Turán number of ${\mathcal{F}}$ is the maximum number of copies of T in an ${\mathcal{F}}$-free graph on n vertices, denoted by $ex(n,T,{\mathcal{F}})$. A linear forest is a graph whose connected components are all paths or isolated vertices. Let ${\mathcal{L}}_{n,k}$ be the family of all linear forests of order n with k edges and $K^*_{s,t}$ be a graph obtained from $K_{s,t}$ by substituting the part of size s with a clique of order s. In this paper, we determine the exact values of $ex(n,K_s,{\mathcal{L}}_{n,k})$ and $ex(n,K^*_{s,t},{\mathcal{L}}_{n,k})$. Also, we study the case of this problem when the “host graph” is bipartite. Denote by $ex_{bip}(n,T,{\mathcal{F}})$ the maximum possible number of copies of T in an ${\mathcal{F}}$-free bipartite graph with each part of size n. We determine the exact value of $ex_{bip}(n,K_{s,t},{\mathcal{L}}_{n,k})$. Our proof is mainly based on the shifting method.