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| ≤ ε n2 and for all 1 ≤ i ≤ k the diameter of the subgraph spanned by Ei is at most d. Using Szemer\'edi's regularity lemma, Polcyn and Ruci\'nski showed that P(n,ε,4) is bounded above by a constant depending only ε. This shows that every dense graph can be partitioned into a small number of ``small worlds'' provided that 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) = (n) is unbounded for ε < 1/4, P(n,ε,3) = (1/ε2) for ε > n-1/2 and P(n,ε,4) = (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, R\"odl, Ruci\'nski, and Szemer\'edi.