  The theory of combinatorial designs has recently been used in order to build
switch-centric data centre networks incorporating a large number of servers, in
comparison with the popular Fat-Tree data centre network. The construction
employed, called the 3-step method, revolves around an appropriately chosen
(but relatively small) bipartite graph and a transversal design. In this paper,
we clarify and extend these recent results. In particular, we prove the
following path diversity results: in a one-to-one context, we prove that in
these data centre networks there are pairwise link-disjoint paths joining all
the servers adjacent to some switch with all the servers adjacent to any other
switch so that we retain control of the path lengths (these results are optimal
in terms of the numbers of paths constructed and we prove that we have a wide
choice of bipartite graph and transversal design to which we can apply the
3-step method); and in a one-to-many context, we prove that there are pairwise
link-disjoint paths from all the servers adjacent to some switch to any
identically-sized collection of target servers where these target servers need
not be adjacent to the same switch (again, we keep control of the path
lengths). Our constructions and analysis are undertaken on bipartite graphs
with the applications to data centre networks being easily derived. Our results
strengthen the overall competitiveness of data centre networks constructed
using the 3-step method, in comparison with Fat-Tree data centre networks, and,
more generally, show the potential of results and methodologies from
combinatorics to data centre network design.
