Path partitions of hypercubes

A path partition of a graph G is a set of vertex-disjoint paths that cover all vertices of G. Given a set P = {{ai, bi}}m i=1 of pairs of distinct vertices of the n-dimensional hypercube Qn, is there a path partition {Pi}m i=1 of Qn such that ai and bi are endvertices of Pi? Caha and Koubek showed that for 6m n, such a path partition exists if and only if the set P is balanced in the sense that it contains the same number of vertices from both classes of bipartition of Qn. In this paper we show that this result holds even for 2m - e < n, where e is the number of pairs of P that form edges of Qn. Moreover, our bound is optimal in the sense that for every n 3, there is a balanced set P in Qn such that 2m - e = n, but no path partition with endvertices prescribed by P exists.

• Type: Journal
• Year: 2008
• Where: IPL
• Authors: Petr Gregor, Tomás Dvorák
• Keywords: IPL 2008, N-dimensional Hypercube Qn, Path Partition, Vertex-disjoint Paths