Cyclically k-partite digraphs and k-kernels

Hortensia Galeana-Sánchez ; César Hernández-Cruz

Discussiones Mathematicae Graph Theory, Tome 31 (2011), p. 63-78 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ${V_i}_{i = 0}^{k-1}$ of V(D) such that every arc in D is a $V_i V_{i+1}-arc$ (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.

Détail

Publié le : 2011-01-01

Zbl 1284.05114

EUDML-ID : urn:eudml:doc:270876

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