Toughness and the existence of k-factors. III
β Scribed by Hikoe Enomoto
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 312 KB
- Volume
- 189
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
β¦ Synopsis
In a paper with the same title (Enomoto et al., 1985) we proved Chv&al's conjecture that ktough graphs have k-factors if they satisfy trivial necessary conditions. In this paper, we introduce a variation of toughness, and prove a stronger result for the existence of l-or 2-factors. This solves a conjecture of Liu and Yu, affirmatively. @
π SIMILAR VOLUMES
## Theorem 2. Let G be a 2-tough graph. Then for any function f : V(G)+ { 1, 2) such that C xsvCcj f (x) in euen, G has an f-factor. Before stating the second main theorem of this paper it is necessary to make the following definition. Let G be a graph and let g and f be two integer-valued functi
## Abstract Degree conditions on the vertices of a __t__βtough graph __G__(1 β¦ __t__ β¦ 2) that ensure the existence of a 2βfactor in __G__ are presented. These conditions are asymptotically best possible for every __t__ Ο΅ [1, 3/2] and for infinitely many __t__ Ο΅ [3/2, 2].
## Abstract Let __k__ be an integer such that β¦, and let __G__ be a connected graph of order __n__ with β¦, __kn__ even, and minimum degree at least __k__. We prove that if __G__ satisfies max(deg(u), deg(v)) β¦ n/2 for each pair of nonadjacent vertices __u, v__ in __G__, then __G__ has a __k__βfacto
In this paper we use Tutte's f-factor theorem and the method of amalgamations to find necessary and sufficient conditions for the existence of a k-factor in the complete multipartite graph K(p(1 ) ..... p(n)), conditions that are reminiscent of the Erd6s-Gallai conditions for the existence of simple
A necessary condition for the existence of a C,-factorization of KZn -F is that k divides 2n. It is known that neither K, -F nor K,, -F admit a C,-factorization. In this paper we show that except for these two cases, the necessary condition is also sufficient.