Mutually independent bipanconnected property of hypercube

A graph is denoted by G with the vertex set V(G) and the edge set E(G). A path P = < v(0), v(1),..., v(m)> is a sequence of adjacent vertices. Two paths with equal length P-1 = < u(1), u(2),..., u(m)> and P-2 = < v(1), v(2),..., v(m)> from a to b are independent if u(1) = v(1) = a, u(m) = v(m) = b, and u(i) not equal v(i) for 2 <= i 6 <= m - 1. Paths with equal length {P-i}(i-1)(n) from a to b are mutually independent if they are pairwisely independent. Let u and v be two distinct vertices of a bipartite graph G, and let l be a positive integer length, d(G)(u, v) <= l <= vertical bar V(G) - 1 vertical bar with (l - d(G)(u, v)) being even. We say that the pair of vertices u, v is ( m, l)- mutually independent bipanconnected if there exist m mutually independent paths {P-i(l)}(i-1)(m) with length l from u to v. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with d(Qn) (u, v) >= n - 1, is (n - 1,l)-mutually independent bipanconnected for every l; d(Qn) (u, v) <= l <= vertical bar V(Q(n))-1 vertical bar with (l - d(Qn) (u, v)) being even. As for d(Qn) (u, v) <= n - 2, it is also (n - 1,l)-mutually independent bipanconnected if l >= d(Qn) (u, v) + 2, and and is only (l,l)-mutually independent bipanconnected if l = d(Qn)(u, v). (C) 2010 Elsevier Inc. All rights reserved.