  Two-level factorial designs are widely used in industrial experiments. For
processes involving \(n\) factors, the construction of designs comprising
\(2^n\) and \(2^{n-p}\) factorials, arranged in blocks of size \(2^q\) is
investigated. The aim is to estimate all main effects and a selected subset of
two-factor interactions. Designs constructed according to minimum aberration
criteria are shown to not necessarily be the most appropriate designs in this
situation. A design construction approach is proposed which exploits known
results on proper vertex colourings in graph theory. Examples are provided to
illustrate the results and construction strategies.
  Particular consideration is given to the special case of designs with blocks
of size four and tables of designs are given for this block size.
