Explicit Bounded-Degree Unique-Neighbor Concentrators

Here we solve an open problem considered by various researchers by presenting the first explicit constructions of an infinite family \( {\user1{\mathcal{F}}} \) of bounded-degree ‘unique-neighbor’ concentrators Γ; i.e., there are strictly positive constants α and ε, such that all Γ = (X,Y,E(Γ)) ∈ \( {\user1{\mathcal{F}}} \) satisfy the following properties. The output-set Y has cardinality \( \frac{{21}} {{22}} \) times that of the input-set X, and for each subset S of X with no more than α|X| vertices, there are at least ε|S| vertices in Y that are adjacent in Γ to exactly one vertex in S. Also, the construction of \( {\user1{\mathcal{F}}} \) is simple to specify, and each \( \Gamma \in {\user1{\mathcal{F}}} \) has fewer than \( \frac{{7{\left| {V{\left( \Gamma \right)}} \right|}}} {2} \) edges. We then modify \( {\user1{\mathcal{F}}} \) to obtain explicit unique-neighbor concentrators of maximum degree 3.

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Correspondence to Michael Capalbo*.

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* Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013.

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Capalbo*, M. Explicit Bounded-Degree Unique-Neighbor Concentrators. Combinatorica 25, 379–391 (2005). https://doi.org/10.1007/s00493-005-0022-x

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Mathematics Subject Classification (2000):

  • 05C35