Excluding an induced star in dense random graphs

Abstract

For fixed $k\geq3$, we study the asymptotic number and typical structure of dense graphs with no induced copy of the star $K_{1,k}$. We solve the associated graphon variational problems both at fixed constant edge density $γ$ and for the conditioned Erdős--Rényi random graph $G(n,p)$ for constant $p$. As consequences, we obtain explicit formulas for the entropy density of induced-$K_{1,k}$-free graphs with $Θ(n^2)$ edges and for the large deviation rate function for the event that $G(n,p)$ is induced-$K_{1,k}$-free. The entropy density exhibits a second-order phase transition at an explicit critical density $γ_k$, while the rate function exhibits a first-order phase transition at a critical parameter $p_k$. We completely characterize the optimizers of both variational problems. Both models have parameter values for which there are infinitely many optimal graphons, but there is always a unique graphon that represents the typical structure in cut metric. We refine the graphon-level results by giving a detailed structural description of both models. For supercritical parameters, each random graph model is the complement of a $(k-1)$-partite graph with high probability. In the subcritical regime of the fixed-density model, the typical structure is the disjoint union of the complement of a $(k-1)$-partite graph, and a sparse remainder. In the subcritical regime of the conditioned Erdős--Rényi random graph, a typical sample has $o(n^2)$ edges.