 
Summary: Sparse balanced partitions and the complexity of subgraph
problems
Noga Alon
DŽaniel Marx
November 14, 2010
Abstract
We consider the problem of partitioning the vertices of a graph with maximum degree
d into k classes V1, . . . , Vk of almost equal sizes in a way that minimizes the number of
pairs (i, j) such that there is an edge between Vi and Vj. We show that there is always
such a partition with O(k22/d
) adjacent pairs and this bound is tight. This problem is
related to questions about the depth of certain graph embeddings, which have been used
in the study of the complexity of subgraph and constraint satisfaction problems.
1 Introduction
If we randomly partition the vertices of a large graph G into a small number k of classes V1,
. . . , Vk of roughly equal sizes, then we expect that every pair (Vi, Vj) of classes is adjacent
(meaning that there is at least one edge with one endpoint in Vi and the other in Vj). This is
true even if the graph G is sparse, for example, if it is a dregular graph for any fixed positive
d. The first question we investigate is whether it is always possible to partition the vertices
in a balanced way into k classes such that the number of pairs of classes that are adjacent
