Sparse balanced partitions and the complexity of subgraph problems

We consider the problem of partitioning the vertices of an n-vertex graph with maximum degree d into k classes V₁;⋯, V_k of size at most ⌊n/k⌋ in a way that minimizes the number of pairs (i,j) for which there is an edge between V_i and V_j. We show that there is always such a partition with O(k^2-2/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.