5-factor-critical Graphs on the Torus

A graph of order n is said to be k-factor-critical for non-negative integer k <= n if the removal of any k vertices results in a graph with a perfect matching. For a k-factor-critical graph of order n, it is called trivial if k = n and non-trivial otherwise. It is known that the toroidal graphs are at most non-trivial 5-factor-critical. Motivated by this, we are to characterize all non-trivial 5-factor-critical graphs on the torus in this paper.

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