DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 18(2) (1998) 197-204
DOI: 10.7151/dmgt.1075

A SUFFICIENT CONDITION FOR THE EXISTENCE OF k-KERNELS IN DIGRAPHS

H. Galeana-Sánchez

Instituto de Matemáticas, UNAM
Ciudad Universitaria, Circuito Exterior
04510 México, D.F., México

H.A. Rincón-Mejía

Departamento de Matemáticas, Facultad de Ciencias
UNAM, Ciudad Universitaria, Circuito Exterior
04510 México, D.F., México

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

[1] C. Berge, Graphs and hypergraphs (North-Holland, Amsterdan, 1973).
[2] P. Duchet, Graphes Noyau-Porfaits, Ann. Discrete Math. 9 (1980) 93-101, doi: 10.1016/S0167-5060(08)70041-4.
[3] P. Duchet, A sufficient condition for a digraph to be kernel-perfect, J. Graph Theory 11 (1987) 81-85, doi: 10.1002/jgt.3190110112.
[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.
[6] M. Kwaśnik, On (k,l)-kernels of exclusive disjunction, cartesian sum and normal product of two directed graphs, Discussiones Math. V (1982) 29-34.

Received 17 November 1997
Revised 10 March 1998