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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1075,
author = {H. Galeana-S\'anchez and H.A. Rinc\'on-Mej\'\i a},
title = {A sufficient condition for the existence of k-kernels in digraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {18},
year = {1998},
pages = {197-204},
zbl = {0928.05032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1075}
}
H. Galeana-Sánchez; H.A. Rincón-Mejía. A sufficient condition for the existence of k-kernels in digraphs. Discussiones Mathematicae Graph Theory, Tome 18 (1998) pp. 197-204. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1075/
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