Disjoint cycles in hypercubes with prescribed vertices in each cycle

[[abstract]]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