k-factors in regular graphs and edge-connectivity

Let l, m, n, r be integers such that 2 <= l <= r, r >= 4 and m, n >= 0. Suppose G is an r-regular l-edge-connected graph and that k is an even integer with m <= k <= r /2. We say that G is an (m, n; k)-factor graph if for each disjoint pair E1, E2 is an element of E(G) with | E-1| = m and | E-2(vertical bar)= n, G has a k-factor F such that E1 is an element of E(F) and E-2 boolean AND E(F) = O. In this note we consider when G is an (m, n; k)-factor graph and characterize those graphs which fail for certain parameters l, m, n, r, k.