  Adaptive mesh refinement is a key component of efficient unstructured
space-time finite element methods. Underlying any adaptive mesh refinement
scheme is, of course, a method for local refinement of simplices. However,
simplex bisection algorithms in dimension greater than three have strict mesh
preconditions which can be hard to satisfy. We prove that certain
four-dimensional simplex space-time meshes can be handled with a relaxed
precondition. Namely, we prove that if a tetrahedral mesh is 4-colorable, then
we can produce a 4D simplex mesh which always satisfies the bisection
precondition. We also briefly discuss strategies to handle tetrahedral meshes
which are not 4-colorable.
