Multi-way expansion constants and partitions of a graph

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In this paper, we consider a relation between $k$-way expansion constant of a finite graph and the expansion constants of subgraphs in a $k$-partition of the graph.

• Using this relation, we show that a sequence of finite graphs which have uniformly bounded $k+1$-way expansion constants and uniformly bounded degrees can be divided into $k$ or less sequences of expanders.
• Furthermore, we prove that such sequence of finite graphs is not coarsely embeddable into any Hilbert space.