Cyclically k-partite digraphs and k-kernels

• Autorzy: Hortensia Galeana-Sánchez , César Hernández-Cruz
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

digraph; kernel; (k,l)-kernel; k-kernel; cyclically k-partite

Kategorie tematyczne

• 05C20: Directed graphs (digraphs), tournaments

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2011
• Tom: 31
• Numer: 1
• Strony: 63-78

Daty

wydano

2011

otrzymano

2009-06-16

poprawiono

2010-04-05

zaakceptowano

2010-04-06

Twórcy

autor

Hortensia Galeana-Sánchez

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

autor

César Hernández-Cruz

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

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Identyfikatory

DOI

10.7151/dmgt.1530