We consider the problem of partitioning the vertices of an n-vertex graph with maximum degree d into k classes V1;⋯, Vk of size at most ⌊n/k⌋ in a way that minimizes the number of pairs (i,j) for which 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.
- Graph partitions
- Subgraph problems