| Summary: | The author proved independently, every 2-tough graph is hamiltonian (3), that if G is a K-tough graph with n vertices and if kn is even then G has a k-factor (4) is best possible by showing that for every positive integer k and for every positive $ varepsilon$ there is a (k-$ varepsilon$)-tough graph G with no k-factor; furthermore, if k is odd then G can be made to have an even number of vertices; this result makes chapter 1 of the present thesis. === Enomoto elaborated further on the subject and gave two results, in (4) and (5) respectively, which are improvements over (4): === (5) provided that k$ vert$G$ vert$ is even and $ vert$G$ vert sbsp{=}{>}$ k + 1, G has a k-factor if, for each set S of vertices, G-S has at most $ vert$S$ vert$/k + 7/8 components; and (6) provided that k$ vert$G$ vert$ is even and $ vert$G$ vert sbsp{=}{>}$ k$ sp{2}$ + 1, G has a k-factor if, for each set S of vertices, G-S has at most $ vert$S$ vert$/k + 1 components. The proofs of (4), (5) and (6) make chapters 2, 3 and 4 respectively.
|
|---|
