One-to-one disjoint path covers on [formula omitted]-ary [formula omitted]-cubes

Abstract

The k -ary n -cube, Q n k , is one of the most popular interconnection networks. Let n ≥ 2 and k ≥ 3 . It is known that Q n k is a nonbipartite (resp. bipartite) graph when k is odd (resp. even). In this paper, we prove that there exist r vertex disjoint paths { P i ∣ 0 ≤ i ≤ r − 1 } between any two distinct vertices u and v of Q n k when k is odd, and there exist r vertex disjoint paths { R i ∣ 0 ≤ i ≤ r − 1 } between any pair of vertices w and b from different partite sets of Q n k when k is even, such that ⋃ i = 0 r − 1 P i or ⋃ i = 0 r − 1 R i covers all vertices of Q n k for 1 ≤ r ≤ 2 n . In other words, we construct the one-to-one r -disjoint path cover of Q n k for any r with 1 ≤ r ≤ 2 n . The result is optimal since any vertex in Q n k has exactly 2 n neighbors.