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Partitions

A partition theorem due to E. Dauber states that, for an edge-transitive hypergraph H = (X, E) {\displaystyle H=(X,E)}, there exists a partition

of the vertex set X {\displaystyle X} such that the subhypergraph H X k {\displaystyle H_{X_{k}}} generated by X k {\displaystyle X_{k}} is transitive for each 1 ≤ k ≤ K {\displaystyle 1\leq k\leq K}, and such that

where r (H) {\displaystyle r(H)} is the rank of H.

As a corollary, an edge-transitive hypergraph that is not vertex-transitive is bicolorable.

Graph partitioning (and in particular, hypergraph partitioning) has many applications to IC design and parallel computing. Efficient and scalable hypergraph partitioning algorithms are also important for processing large scale hypergraphs in machine learning tasks.