The Splitting Number of the 4-Cube

The splitting number of a graph G consists in the smallest positive integer k 0, such that a planar graph can be obtained from G by k splitting operations, such operation replaces v by two nonadjacent vertices v1 and v2, and attaches the neighbors of v either to v1 or to v2. One of the most useful graphs in computer science is the n{cube. Dean and Richter devoted an article to proving that the minimum number of crossings in an optimum drawing of the 4{cube is 8, but no results about splitting number of nonplanar n{cubes are known. In this note we give a proof that the splitting number of the 4{cube is 4. In addition, we give the lower bound 2 n?2 for the splitting number of the n{cube. Because it is known that the splitting number of the n{cube is O(2 n), our result implies that the splitting number of the n-cube is (2 n).