Discussiones
Mathematicae Graph Theory 18(2) (1998) 197-204
DOI: https://doi.org/10.7151/dmgt.1075
A SUFFICIENT CONDITION FOR THE EXISTENCE OF k-KERNELS IN DIGRAPHS
H. Galeana-Sánchez Instituto de Matemáticas, UNAM |
H.A. Rincón-Mejía Departamento de Matemáticas, Facultad de Ciencias |
Abstract
In this paper, we prove the following sufficient condition for the existence of k-kernels in digraphs: Let D be a digraph whose asymmetrical part is strongly conneted and such that every directed triangle has at least two symmetrical arcs. If every directed cycle γ of D with l(γ) ≢ 0 (mod k), k ≥ 2 satisfies at least one of the following properties: (a) γ has two symmetrical arcs, (b) γ has four short chords. Then D has a k-kernel.
This result generalizes some previous results on the existence of kernels and k-kernels in digraphs. In particular, it generalizes the following Theorem of M. Kwaśnik [5]: Let D be a strongly connected digraph, if every directed cycle of D has length ≡ 0 (mod k), k ≥ 2. Then D has a k-kernel.
Keywords: digraph, kernel, k-kernel.
1991 Mathematics Subject Classification: 05C20.
References
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[4] | H. Galeana-Sánchez, On the existence of kernels and k-kernels in directed graphs, Discrete Math. 110 (1992) 251-255, doi: 10.1016/0012-365X(92)90713-P. |
[5] | M. Kwaśnik, The generalization of Richardson theorem, Discussiones Math. IV (1981) 11-14. |
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Received 17 November 1997
Revised 10 March 1998
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