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Berge girth at least three forces linearity
Statement
If a finite hypergraph has Berge girth at least three, then it is linear: any two distinct hyperedges have at most one common vertex.
Statement references
Facts & Assumptions
By Berge cycles, incidence graphs and girth, a Berge cycle of length two consists of two distinct hyperedges and two distinct vertices , each belonging to both hyperedges. Berge girth is the least cycle length, or if there is no cycle, and linearity means that any two distinct hyperedges have at most one common vertex.
Proof
Given: A finite hypergraph of Berge girth at least three.
Goal: The hypergraph is linear.
If two distinct hyperedges had two distinct common vertices, those edges and vertices would form a Berge cycle of length two: each vertex lies in the intersection of the two edges, as required for both cyclic indices.
Such a cycle would make the Berge girth at most two, contrary to the given lower bound. Hence no two distinct hyperedges have two distinct common vertices, so the hypergraph is linear.
Depends on
Dependency tree · 2 levels
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