9 research outputs found
Bounded Degree Spanners of the Hypercube
In this short note we study two questions about the existence of subgraphs of
the hypercube with certain properties. The first question, due to
Erd\H{o}s--Hamburger--Pippert--Weakley, asks whether there exists a bounded
degree subgraph of which has diameter . We answer this question by
giving an explicit construction of such a subgraph with maximum degree at most
120.
The second problem concerns properties of -additive spanners of the
hypercube, that is, subgraphs of in which the distance between any two
vertices is at most larger than in . Denoting by
the minimum possible maximum degree of a -additive
spanner of , Arizumi--Hamburger--Kostochka showed that
We improve their upper bound by showing that
where the
last term denotes a -fold iterated logarithm.Comment: 10 page