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**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.

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.) |

### Citations:

Zbl 0256.05122
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\textit{H. Enomoto} et al., J. Graph Theory 9, No. 1, 87--95 (1985; Zbl 0598.05054)

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### 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[4] | Nearly k-tough graphs with no k-factor, unpublished.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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