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 A(1), A(2) ,..., A(r) of G with vertical bar A(1) boolean OR A(2) boolean OR ... boolean OR A(r)vertical bar <= t, there exist r disjoint cycles C-1, C-2,...,Cr of G such that C-1 boolean OR C-2 boolean OR ... C-r spans G, and C-t contains A(i) for every i. In this paper, we prove that the n-dimensional hypercube Q(n) is spanning 2-cyclable of order n - 1 for n >= 3. Moreover, Q(n) 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 Q(n) is n - for n >= 3. (C) 2013 Elsevier B.V. All rights reserved.