Toughness and the existence of fractional k-factors of graphs Original Research Article

The toughness of a graph G, image, is defined as image where image denotes the number of components of image or image 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 image and has a fractional k-factor if image where image. Furthermore, we show that both results are best possible in some sense.