Decompositions into subgraphs of small diameter

TL;DR

This paper studies how to decompose graphs into a small number of low diameter subgraphs, providing tight bounds on the number needed based on density and minimum degree, and extends results to hypergraphs.

Contribution

It determines the asymptotic bounds for the number of low diameter subgraphs needed for various densities, improving and extending previous results, including hypergraph cases.

Findings

P(n,ε,2) = Θ(n) for ε < 1/4

P(n,ε,3) = Θ(1/ε^2) for ε > n^{-1/2}

P(n,ε,4) = Θ(1/ε) for ε > n^{-1}

Abstract

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.