Generalized Turán Number for Edge Blow-up of Paths and Cycles

Abstract

Given a graph H and an integer p ≥ 2, the edge blow-up graph H^p+1 of H is the graph obtained by replacing each edge in H with a clique of order p + 1, where the new vertices of the cliques are all distinct. The generalized Turán number ex(n, K_m, F) denote the maximum number of copies of K_m in an n-vertex F-free graph. Let C_t and P_t denote the cycle and path with t vertices, respectively. In this paper, we obtain the generalized Turán numbers ex(n, K_m, P ^p+1_t ), ex(n, K_m, C ^p+1_t ) and characterize the unique graph for P ^p+1_t and C ^p+1_t respectively, when t ≥ 3, p ≥ m ≥ 3 and n is sufficiently large.