Disjoint cycles in hypercubes with prescribed vertices in each cycle

A graph G is spanning r-cyclable of order t if for any r nonempty mutually disjoint vertex subsets A1, A2, . . . , Ar of G with |A1 ∪ A2 ∪ · · · ∪ Ar | ≤ t , there exist r disjoint cycles C1, C2, . . . , Cr of G such that C1 ∪ C2 ∪ · · · ∪ Cr spans G, and Ci contains Ai for every i. In this paper, we prove that the n-dimensional hypercube Qn is spanning 2-cyclable of order n − 1 for n ≥ 3. Moreover, Qn is spanning k-cyclable of order k if k ≤ n − 1 for n ≥ 2. The spanning r-cyclability of a graph G is the maximum integer t such that G is spanning r-cyclable of order k for k = r, r + 1, . . . , t but is not spanning r-cyclable of order t + 1. We also show that the spanning 2-cyclability of Qn is n − 1 for n ≥ 3. © 2013 Elsevier B.V. All rights reserved.