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

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CYCLICALLY k-PARTITE DIGRAPHS AND k-KERNELS

Hortensia Galeana-Sánchez and César Hernández-Cruz

Instituto de Matemáticas
Universidad Nacional Autónoma de México
Ciudad Universitaria, México, D.F., C.P. 04510, México
e-mail: hgaleana@matem.unam.mx
e-mail: cesar@matem.unam.mx

Abstract

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.

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