A feature of minimizing images of submodular binary Markov random field (MRF) energies is introduced. Using this novel feature, the collection of minimizing images of levels of higher order, monotonically levelable multilabel MRF energies is shown to constitute a monotone collection. This implies that these minimizing binary images can be combined to give minimizing images of the multilabel MRF energies. Thanks to the graph cuts framework, the mentioned class of binary MRF energies is known to be minimized by maximum flow computations on appropriately constructed graphs. With the aid of these developments an exact and efficient algorithm to minimize monotonically levelable multilabel MRF energies of any order, which is composed of a series of maximum flow computations, is proposed and an application of the proposed algorithm to image denoising is given.