Decompositions into Subgraphs of Small Diameter

We investigate decompositions of a graph into a small number of low-diameter subgraphs. Let P(n,epsilon,d) be the smallest k such that every graph G =(V, E) on n vertices has an edge partition E=E(0)boolean OR E(1)boolean OR...boolean OR E(k) such that vertical bar E(0)vertical bar <= epsilon n(2), and for all 1 <= i <= k the diameter of the subgraph spanned by E(i) is at most d. Using Szemeredi's regularity lemma, Polcyn and Rucinski showed that P(n,epsilon,4) is bounded above by a constant depending only on epsilon. 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,epsilon,d) within an absolute constant factor, showing that P(n,epsilon,2) = Theta(n) is unbounded for epsilon < 1/4, P(n,epsilon,3) = Theta(1/epsilon(2)) for epsilon > n(-1/2) and P(n,epsilon,4) = Theta(1/epsilon) for epsilon > 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, Rodl, Rucinski and Szemeredi.