Two spanning disjoint paths with required length in generalized hypercubes

Given two pairs < u, v > and < x, y > of vertices of a graph G = (V, E) and two integers l(1) and l(2) with l(1) + l(2) = vertical bar V(G)vertical bar - 2, G is said to be satisfying the 2RP-property if there exist two disjoint paths P-1 and P-2 such that (1) P-1 is a path joining u to v with l(P-1) = l(1), (2) P-2 is a path joining x to y with l(P-2) = l(2), and (3) P-1 boolean OR P-2 spans G, where l(P) denotes the length of path P. In this paper, we show that an r-dimensional generalized hypercube, denoted by G(m(r), m(r-1), ... , m(1)), satisfies the 2RP-property except some special conditions, where m(i) >= 4 for all 1 <= i <= r. (C) 2013 Elsevier B.V. All rights reserved.