Sparse Balanced Partitions and the Complexity of Subgraph Problems

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(k2−2/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.