## Toughness and the existence of k-factors.(English)Zbl 0598.05054

Finite undirected connected graphs G without loops and multiple edges are considered. If $$S\subset V(G)$$, then by G-S denote the subgraph of G induced by the set V(G)-S and by w(G-S) the number of components of G-S.
A graph G is t-tough if the implication $$w(G-S)>1\Rightarrow | S| \geq t.w(G-S)$$ holds for any $$S\subset V(G)$$. The toughness of a graph was introduced by V. Chvátal who made also the following conjecture [Tough graphs and Hamiltonian circuits, Discrete Math. 5, 215-228 (1973; Zbl 0256.05122)]: If G is k-tough, $$| G| \geq k+1$$ and $$k| G|$$ is even, then G has a k-factor.
The main result of the present paper is the proof of this conjecture (Theorem 1). Further, the result is shown to be sharp: (Theorem 3): For $$k\geq 1$$ and any positive $$\epsilon$$, there exists a (k-$$\epsilon)$$-tough graph with $$k| G|$$ even, $$| G| \geq k+1$$ which has no k- factor.
Reviewer: St.Znám

### MSC:

 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)

### Keywords:

toughness; k-factor

Zbl 0256.05122
Full Text:

### References:

 [1] , and , Graphs & Digraphs. Prindle, Weber & Schmidt, Massachusetts (1979). [2] and , Graph Theory with Applications. Macmillan, London (1976). · Zbl 1226.05083 [3] Chvátal, Discrete Math. 5 pp 215– (1973) [4] Nearly k-tough graphs with no k-factor, unpublished. [5] Tutte, Canad. J. Math. 4 pp 314– (1952) · Zbl 0049.24202
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