Decompositions into subgraphs of small diameter

Jacob Fox, Benny Sudakov

Published 2009-06-18Version 1

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 \cup E_1 \cup ... \cup E_k such that |E_0| \leq \epsilon n^2 and for all 1 \leq i \leq k the diameter of the subgraph spanned by E_i is at most d. Using Szemer\'edi's regularity lemma, Polcyn and Ruci\'nski showed that P(n,\epsilon,4) is bounded above by a constant depending only \epsilon. 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,\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, R\"odl, Ruci\'nski, and Szemer\'edi.