Abstract: Let G be a connected undirected simple graph. If the remaining subgraph still has a fractional perfect matching after deleting any k vertices of G, the graph G is said to be fractional k-factor-critical. This paper gives sufficient conditions in terms of the toughness and the degree sum for a graph G to be fractional k-factor-critical. In some sense, we prove that the conditions are best possible. Besides, we give a necessary and sufficient condition in terms of the fractional matching number for a graph G  to be fractional k-factor-critical.

Key words: fractional perfect matching, fractional k-factor-critical, toughness, fractional matching number