ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 31(1) (2011) 63-78
DOI: 10.7151/dmgt.1530


Hortensia Galeana-Sánchez and César Hernández-Cruz

Instituto de Matemáticas
Universidad Nacional Autónoma de México
Ciudad Universitaria, México, D.F., C.P. 04510, México



Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively.

A (k,l)-kernel N of D is a k-independent set of vertices (if u,v ∈ N then d(u,v) ≥ k) and l-absorbent (if u ∈ V(D)−N then there exists v ∈ N such that d(u,v) ≤ l). A k-kernel is a (k,k−1)-kernel. A digraph D is cyclically k-partite if there exists a partition {Vi}i = 0k−1 of V(D) such that every arc in D is a Vi Vi+1-arc (mod k). We give a characterization for an unilateral digraph to be cyclically k-partite through the lengths of directed cycles and directed cycles with one obstruction, in addition we prove that such digraphs always have a k-kernel. A study of some structural properties of cyclically k-partite digraphs is made which bring interesting consequences, e.g., sufficient conditions for a digraph to have k-kernel; a generalization of the well known and important theorem that states if every cycle of a graph G has even length, then G is bipartite (cyclically 2-partite), we prove that if every cycle of a graph G has length ≡ 0 (mod k) then G is cyclically k-partite; and a generalization of another well known result about bipartite digraphs, a strong digraph D is bipartite if and only if every directed cycle has even length, we prove that an unilateral digraph D is bipartite if and only if every directed cycle with at most one obstruction has even length.

Keywords: digraph, kernel, (k,l)-kernel, k-kernel, cyclically k-partite.

2010 Mathematics Subject Classification: 05C20.


[1] J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications (Springer-Verlag, 2002).
[2] C. Berge, Graphs (North-Holland, Amsterdam, New York, 1985).
[3] C. Berge and P. Duchet, Recent problems and results about kernels in directed graphs, Discrete Math. 86 (1990) 27-31, doi: 10.1016/0012-365X(90)90346-J.
[4] R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory (Encyclopedia of Mathematics and its Applications) (Cambridge University Press, 1991).
[5] R. Diestel, Graph Theory 3rd Edition (Springer-Verlag, Heidelberg, New York, 2005).
[6] H. Galeana-Sánchez, On the existence of kernels and h-kernels in directed graphs, Discrete Math. 110 (1992) 251-255, doi: 10.1016/0012-365X(92)90713-P.
[7] M. Kucharska and M. Kwaśnik, On (k,l)-kernels of special superdigraphs of Pm and Cm, Discuss. Math. Graph Theory 21 (2001) 95-109, doi: 10.7151/dmgt.1135.
[8] M. Kwaśnik, On (k,l)-kernels on graphs and their products, Doctoral dissertation, Technical University of Wroc aw, Wroc aw, 1980.
[9] M. Kwaśnik, The generalizaton of Richardson's theorem, Discuss. Math. 4 (1981) 11-14.
[10] M. Richardson, On weakly ordered systems, Bull. Amer. Math. Soc. 52 (1946) 113-116, doi: 10.1090/S0002-9904-1946-08518-3.
[11] A. Sánchez-Flores, A counterexample to a generalization of Richardson's theorem, Discrete Math. 65 (1987) 319-320.
[12] W. Szumny, A. Włoch and I. Włoch, On (k,l)-kernels in D-join of digraphs, Discuss. Math. Graph Theory 27 (2007) 457-470, doi: 10.7151/dmgt.1373.
[13] W. Szumny, A. Włoch and I. Włoch, On the existence and on the number of (k,l)-kernels in the lexicographic product of graphs, Discrete Math. 308 (2008) 4616-4624, doi: 10.1016/j.disc.2007.08.078.
[14] J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, Princeton, 1953).
[15] A. Włoch and I. Włoch, On (k,l)-kernels in generalized products, Discrete Math. 164 (1997) 295-301, doi: 10.1016/S0012-365X(96)00064-7.

Received 16 June 2009
Revised 5 April 2010
Accepted 6 April 2010