Decompositions into Subgraphs of Small Diameter

Abstract

We investigate decompositions of a graph into a small number of low-diameter subgraphs. Let P(n, ε, d) be the smallest k such that every graph G = (V, E) on n vertices has an edge partition E = E0 ∪ E1 ∪ ⋅⋅⋅ ∪ Ek such that |E0| ≤ εn2, and for all 1 ≤ i ≤ k the diameter of the subgraph spanned by Ei is at most d. Using Szemerédi's regularity lemma, Polcyn and Ruciński showed that P(n, ε, 4) is bounded above by a constant depending only on ε. This shows that every dense graph can be partitioned into a small number of ‘small worlds’ provided that a few edges can be ignored. Improving on their result, we determine P(n, ε, d) within an absolute constant factor, showing that P(n, ε, 2) = Θ(n) is unbounded for ε < 1/4, P(n, ε, 3) = Θ(1/ε2) for ε > n−1/2 and P(n, ε, 4) = Θ(1/ε) for ε > n−1. We also prove that if G has large minimum degree, all the edges of G can be covered by a small number of low-diameter subgraphs. Finally, we extend some of these results to hypergraphs, improving earlier work of Polcyn, Rödl, Ruciński and Szemerédi.