Toughness and the Existence of Fractional 3-Factor

Let G be a graph, a variation of Toughness is defined as τ (G) =min{ ( ) 1Gω G S ,S V(G), ω(G S) ≥ 2} if G is not complete, and τ (G) =∞ if G is complete. One conjectureconcern toughness and exist of fractional k-factor is that: Let G be a graph, if τ (G)>k for integerk ≥ 2, then G has fractional k-factor. This conjecute has proof true for k=2. In this paper, we provethat G has fraction 3-factor if τ (G) >3.