The Generalized Turán Number of Spanning Linear Forests

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