Geodesic
A sufficient condition for the existence of k-kernels in digraphs
Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 2, pp. 197-204

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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.
Keywords: digraph, kernel, k-kernel 
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Galeana-Sánchez, H.; Rincón-Mejía, H. A sufficient condition for the existence of k-kernels in digraphs. Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 2, pp. 197-204. http://geodesic.mathdoc.fr/item/DMGT_1998_18_2_a5/