ON THE EXISTENCE OF DISJOINT SPANNING PATHS IN FAULTY HYPERCUBES

Abstract

Assume that n is a positive integer with n ≥ 4 and F is a subset of the edges of the hypercube Qn with |F| ≤ n-4. Let u , x be two distinct white vertices of Qn and v , y be two distinct black vertices of Qn, where black and white refer to the two parts of the bipartition of Qn. Let l1 and l2 be odd integers, where l1 ≥ dQn-F( u , v ), l2 ≥ dQn-F( x , y ), and l1 + l2 = 2n - 2. Moreover, let l3 and l4 be even integers, where l3 ≥ dQn-F( u , x ), l4 ≥ dQn-F( v , y ), and l3+l4 = 2n - 2. In this paper, we prove that there are two disjoint paths P1 and P2 such that (1) P1 is a path joining u to v with length l(P1) = l1, (2) P2 is a path joining x to y with l(P2) = l2, and (3) P1 ∪ P2 spans Qn - F. Moreover, there are two disjoint paths P3 and P4 such that (1) P3 is a path joining u to x with l(P3) = l3, (2) P4 is a path joining v to y with l(P4) = l4, and (3) P3 ∪ P4 spans Qn - F except the following cases: (a) l3 = 2 with dQn-F( u , x ) = 2 and dQn-F-{ v , y }( u , x ) > 2, and (b) l4 = 2 with dQn-F( v , y ) = 2 and dQn-F-{ u , x }( v , y ) > 2.