# A sufficient condition for the existence of k-kernels in digraphs

H. Galeana-Sánchez; H.A. Rincón-Mejía

Discussiones Mathematicae Graph Theory (1998)

- Volume: 18, Issue: 2, page 197-204
- ISSN: 2083-5892

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topH. Galeana-Sánchez, and H.A. Rincón-Mejía. "A sufficient condition for the existence of k-kernels in digraphs." Discussiones Mathematicae Graph Theory 18.2 (1998): 197-204. <http://eudml.org/doc/270530>.

@article{H1998,

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.
},

author = {H. Galeana-Sánchez, H.A. Rincón-Mejía},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {digraph; kernel; k-kernel; -kernel},

language = {eng},

number = {2},

pages = {197-204},

title = {A sufficient condition for the existence of k-kernels in digraphs},

url = {http://eudml.org/doc/270530},

volume = {18},

year = {1998},

}

TY - JOUR

AU - H. Galeana-Sánchez

AU - H.A. Rincón-Mejía

TI - A sufficient condition for the existence of k-kernels in digraphs

JO - Discussiones Mathematicae Graph Theory

PY - 1998

VL - 18

IS - 2

SP - 197

EP - 204

AB -
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.

LA - eng

KW - digraph; kernel; k-kernel; -kernel

UR - http://eudml.org/doc/270530

ER -

## References

top- [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. Zbl0607.05036
- [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. Zbl0509.05048
- [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. Zbl0508.05038

## Citations in EuDML Documents

top- Pietra Delgado-Escalante, Hortensia Galeana-Sánchez, Kernels and cycles' subdivisions in arc-colored tournaments
- Pietra Delgado-Escalante, Hortensia Galeana-Sánchez, On monochromatic paths and bicolored subdigraphs in arc-colored tournaments
- H. Galeana-Sánchez, C. Hernández-Cruz, On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey

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