Edge-disjoint induced subgraphs with given minimum degree

Abstract

Let h be a given positive integer. For a graph with n vertices and m edges, what is the maximum number of pairwise edge-disjoint induced subgraphs, each having minimum degree at least h? There are examples for which this number is O(m2=n2). We prove that this bound is achievable for all graphs with polynomially many edges. For all ε> 0, if m > n1+ε, then there are always Ω(m2=n2) pairwise edge-disjoint induced subgraphs, each having minimum degree at least h. Furthermore, any two subgraphs intersect in an independent set of size at most 1 + O(n3=m2), which is shown to be asymptotically optimal.