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.