Decompositions into subgraphs of small diameter

  • Authors:

  • Jacob Fox;Benny Sudakov

  • Affiliations:

  • Department of mathematics, princeton, princeton, nj, usa (e-mail: jacobfox@math.princeton.edu);Department of mathematics, ucla, los angeles, ca 90095, usa (e-mail: bsudakov@math.ucla.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2010

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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 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.