Toughness and the existence of fractional \(k\)-factors of graphs (Q2477400)

In this paper, the relationship between the toughness and the existence of fractional \(k\)-factor of an undirected graph is discussed. Firstly, the authors prove that \(G\) has a fractional perfect matching if toughness of a connected graph is at least 1. Secondly, for any integer \(k \geq 2\), if the toughness of a graph is at least \((k-1)/k\) then the graph is shown to have a fractional \(k\)-factor. The results are sharp for both cases.