A Localized Approach to Generalized Turán Problems

Generalized Turán problems ask for the maximum number of copies of a graph H in an n-vertex, F-free graph, denoted by $ex(n,H,F)$. We show how to extend the new, localized approach of Bradač, Malec, and Tompkins to generalized Turán problems. We weight the copies of H (typically taking H=Kₜ), instead of the edges, based on the size of the largest clique, path, or star containing the vertices of the copy of H, and in each case prove a tight upper bound on the sum of the weights. The generalized edge Turán number $mex(m,H,F)$ is the maximum number of copies of a graph H in an m-edge, F-free graph. A consequence of our new localized theorems is an asymptotic determination of ex(n,H,K1,r) for every H having at least one dominating vertex and mex(m,H,K1,r) for every H having at least two dominating vertices.