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.

翻译：适应性网格改进是高效无结构的空间-时间限制元素方法的关键组成部分。 当然, 任何适应性网格改进计划的基础, 当然是局部改进软体的方法。 但是, 尺寸大于3的简单二分算算法有严格的网格先决条件, 很难满足。 我们证明某些四维的简单空格- 时空网列可以用一个宽松的先决条件来处理。 也就是说, 我们证明如果四面网列是4色的, 那么我们可以产生一个 4D 简单的网格, 总是满足两面的前提条件。 我们还简短地讨论处理四面网列的策略, 而不是4色的。