The splitting number of the 4-cube

Abstract

The splitting number of a graph G is the smallest 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 vl and v2, and attaches the neighbors of v either to vl or to v2. The n-cube has a distinguished place in Computer Science. 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 other 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 2n-2for the splitting number of the n-cube. It is known that the splitting number of the n-cube is O(2n), thus our result implies that the splitting number of the n-cube is Ø(2n).