Toughness and the existence of fractional k-factors of graphs

The toughness of a graph G, t(G), is defined as t(G)=min{|S|/@w(G-S)|[email protected]?V(G),@w(G-S)>1} where @w(G-S) denotes the number of components of G-S or t(G)=+~ if G is a complete graph. Much work has been contributed to the relations between toughness and the existence of factors of a graph. In this paper, we consider the relationship between the toughness and the existence of fractional k-factors. It is proved that a graph G has a fractional 1-factor if t(G)>=1 and has a fractional k-factor if t(G)>=k-1/k where k>=2. Furthermore, we show that both results are best possible in some sense.