Mixed-cycle-V-super magic decomposition of complete bipartite graphs

An [Formula: see text]-magic labeling in an [Formula: see text]-decomposable graph [Formula: see text] is a bijection [Formula: see text] such that for every copy [Formula: see text] in the decomposition, [Formula: see text] is constant. [Formula: see text] is said to be [Formula: see text]-[Formula: see text]-super magic if [Formula: see text]. A family of subgraphs [Formula: see text] of [Formula: see text] is a mixed-cycle-decomposition of [Formula: see text] if every subgraph [Formula: see text] is isomorphic to some cycle [Formula: see text], for [Formula: see text], [Formula: see text] for [Formula: see text] and [Formula: see text]. In this paper, we prove that [Formula: see text] is mixed-cycle-[Formula: see text]-super magic decomposable where [Formula: see text], [Formula: see text], with the help of the results found in [C.-C. Chou, C.-M. Fu and W.-C. Huang, Decomposition of [Formula: see text] into short cycles, Discrete Math. 197/198 (1999) 195–203].