mixed cycle-e-super magic decomposition of complete bipartite graphs

an h-magic labeling in a h-decomposable graph g is a bijection f : v (g) ∪ e(g) → {1, 2, ..., p + q} such that for every copy h in the decomposition, σνεv(h) f(v) +  σeεe(h) f(e) is constant. f is said to be h-e-super magic if f(e(g)) = {1, 2, · · · , q}. a family of subgraphs h1,h2, · · · ,hh of g is a mixed cycle-decomposition of g if every subgraph hi is isomorphic to some cycle ck, for k ≥ 3, e(hi) ∩ e(hj) = ∅ for i ≠ j and ∪hi=1e(hi) = e(g). in this paper, we prove that k2m,2n is mixed cycle-e-super magic decomposable where m ≥ 2, n ≥ 3, with the help of the results found in [1].