The t-stability number of a random graph

Abstract

Given a graph G = (V,E), a vertex subset S ⊆ V is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number αt(G) of G is the maximum order of a t-stable set in G. The theme of this paper is the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n → 8, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression for the expected number of t-stable sets of order k. We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most t.