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

• Autorzy: H. Galeana-Sánchez , H.A. Rincón-Mejía
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

digraph; kernel; k-kernel

Kategorie tematyczne

• 05C20: Directed graphs (digraphs), tournaments

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 1998
• Tom: 18
• Numer: 2
• Strony: 197-204

Daty

wydano

1998

otrzymano

1997-11-17

poprawiono

1998-03-10

Twórcy

autor

H. Galeana-Sánchez

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

autor

H.A. Rincón-Mejía

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

Bibliografia

• [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.

Identyfikatory

DOI

10.7151/dmgt.1075