Further Results on the Generalized Turán Number of Spanning Linear Forests

Abstract

A linear forest is a graph consisting of vertex disjoint paths. Let l(G) denote the maximum size of linear forests in G. Denote by $\delta (G)$ the minimum degree of G. Recently, Duan, Wang and Yang gave an upper bound on the number of 3-cliques in n-vertex graphs with $l(G)=k-1$ and $\delta (G)=\delta .$ Duan et al. gave an upper bound $h_s(n,\alpha ',\delta )$ on the number of s-cliques in n-vertex graphs with prescribed matching number $\alpha '$ and minimum degree $\delta .$ But in some cases, these two upper bounds are not obtained by the graph with minimum degree $\delta .$ For example, $h_2(15,7,3)=77$ is attained by a unique graph of minimum degree 7,  not 3. Motivated by these works, we give sharp results about this problem. We determine the maximum number of s-cliques in n-vertex graphs with $l(G)=k-1$ and $\delta (G)=\delta .$ As a corollary of our main results, we determine the maximum number of s-cliques in n-vertex graphs with given matching number and minimum degree. Moreover, we also determine the maximum number of copies of $K_{r_1,r_2},$ the complete bipartite graph with class sizes $r_1$ and $r_2,$ in n-vertex graphs with $l(G)=k-1$ and $\delta (G)=\delta .$