The assignment problem, which is the most degenerate form of the transportation problem, and therefore the most likely to cycle, is investigated by means of linear graphs. It is proved that cycling is impossible in the 3 × 3 case, but possible in the 4 × 4 case. For the 5 × 5 case, it is shown both that there is a cycle independent of the one found for the 4 × 4 case and also that the cycle found in the 4 × 4 case can exist in the 5 × 5 case. Numerical examples are given.
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Betty Jane Gassner (1964) studied this question.