Toughness and the existence of fractional k-factors of graphs

The toughness of a graph G  , t(G)t(G), is defined as t(G)=min{|S|/ω(G-S)|S⊆V(G),ω(G-S)>1}t(G)=min{|S|/ω(G-S)|S⊆V(G),ω(G-S)>1} where ω(G-S)ω(G-S) denotes the number of components of G-SG-S or t(G)=+∞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)⩾1t(G)⩾1 and has a fractional k  -factor if t(G)⩾k-1/kt(G)⩾k-1/k where k⩾2k⩾2. Furthermore, we show that both results are best possible in some sense.