The feasible region of induced graphs

Abstract

The feasible region Ωind(F) of a graph F is the collection of points (x,y) in the unit square such that there exists a sequence of graphs whose edge densities approach x and whose induced F-densities approach y. A complete description of Ωind(F) is not known for any F with at least four vertices that is not a clique or an independent set. The feasible region provides a lot of combinatorial information about F. For example, the supremum of y over all (x,y)∈Ωind(F) is the inducibility of F and Ωind(Kr) yields the Kruskal-Katona and clique density theorems.We begin a systematic study of Ωind(F) by proving some general statements about the shape of Ωind(F) and giving results for some specific graphs F. Many of our theorems apply to the more general setting of quantum graphs. For example, we prove a bound for quantum graphs that generalizes an old result of Bollobás for the number of cliques in a graph with given edge density. We also consider the problems of determining Ωind(F) when F=Kr−, F is a star, or F is a complete bipartite graph. In the case of Kr− our results sharpen those predicted by the edge-statistics conjecture of Alon et al. while also extending a theorem of Hirst for K4− that was proved using computer aided techniques and flag algebras. The case of the 4-cycle seems particularly interesting and we conjecture that Ωind(C4) is determined by the solution to the triangle density problem, which has been solved by Razborov.