Two Spanning Disjoint Paths with Required Length in Augmented Cubes

Abstract

In this article, we introduce 2RP-property in the augmented cube AQn: Let {u, v, x, y} be any four distinct vertices of AQn. Let l1 and l2 be two integers with l1 ≥ dAQn(u, v), l2 ≥ dAQn(x, y), and l1+l2 = 2n−2. Then there exist two disjoint paths P1 and P2 such that (1) P1 is a path joining u and v with l(P1) = l1, (2) P2 is a path joining x and y with l(P2) = l2, and (3) P1 ∪ P2 spans AQn except some special conditions.