Kernels in edge-colored digraphs

Galeana‐Sánchez, Hortensia

doi:10.1016/s0012-365x(97)00162-3

Cited by 38 publications

(19 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let D be a digraph, an m-coloration of D is called {C 3 , C 4 }-free m-coloration of D if it is an m-coloration of D such that every directed cycle of length at most 4 is quasi-monochromatic. Denote by F the class of digraphs such that the closure of every {C 3 , C 4 }-free m-coloration has a kernel; in [2] it is proved that every tournament belongs to F and in [3] it is proved that the digraph obtained from a tournament by the deletion of a single arc belongs to F.…”

Section: Antecedentesmentioning

confidence: 99%

Kernels in the closure of coloured digraphs

Galeana‐Sánchez¹,

García-Ruvalcaba²

2000

Discuss. Math. Graph Theory

Self Cite

View full text Add to dashboard Cite

Let D be a digraph with V (D) and A(D) the sets of vertices and arcs of D, respectively. A kernel of D is a set I ⊂ V (D) such that no arc of D joins two vertices of I and for each x ∈ V (D) \ I there is a vertex y ∈ I such that (x, y) ∈ A(D). A digraph is kernel-perfect if every non-empty induced subdigraph of D has a kernel. If D is edge coloured, we define the closure ξ(D) of D the multidigraph with V (ξ(D)) = V (D) and A (ξ(D)) = i {(u, v) with colour i : there exists a monochromatic path of colour i from the vertex u to the vertex v contained in D}. Let T 3 and C 3 denote the transitive tournament of order 3 and the 3-cycle, respectively, both of whose arcs are coloured with 3 different colours. In this paper, we survey sufficient conditions for the existence of kernels in the closure of edge coloured digraphs, also we prove that if D is obtained from an edge coloured tournament by deleting one arc and D does not contain T 3 or C 3 , then ξ(D) is a kernel-perfect digraph.

show abstract

Section: Antecedentesmentioning

confidence: 99%

Kernels in the closure of coloured digraphs

Galeana‐Sánchez¹,

García-Ruvalcaba²

2000

Discuss. Math. Graph Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…Besides, Galeana-Sánchez and Rojas-Monroythe [15] showed in 2004 that every arc-colored bipartite tournament with all 4-cycles monochromatic has an MP-kernel. For more results on MP-kernels, we refer to the survey paper [11] by Galeana-Sánchez.…”

Section: Introductionmentioning

confidence: 99%

Kernels by properly colored paths in arc-colored digraphs

Bai

Fujita

Zhang

2018

Discrete Mathematics

View full text Add to dashboard Cite

A kernel by properly colored paths of an arc-colored digraph D is a set S of vertices of D such that (i) no two vertices of S are connected by a properly colored directed path in D, and (ii) every vertex outside S can reach S by a properly colored directed path in D. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for unicyclic digraphs, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.

show abstract

“…In the coloured case, independence and absorbency are only required by means of monochromatic paths, hence these sets are called kernels by monochromatic paths. The existence of such kernels is widely studied, see [7]- [12], [14]. The case when K is an absorbing set but not necessarily independent by monochromatic paths is also of interest.…”

Section: Previous Workmentioning

confidence: 99%

King-Serf Duo by Monochromatic Paths in $k$-Edge-Coloured Tournaments

Bérczi

Joó

2017

Electron. J. Combin.

View full text Add to dashboard Cite

An open conjecture of Erdős states that for every positive integer k there is a (least) positive integer f (k) so that whenever a tournament has its edges colored with k colors, there exists a set S of at most f (k) vertices so that every vertex has a monochromatic path to some point in S. We consider a related question and show that for every (finite or infinite) cardinal κ > 0 there is a cardinal λ κ such that in every κ-edge-coloured tournament there exist disjoint vertex sets K, S with total size at most λ κ so that every vertex v has a monochromatic path of length at most two from K to v or from v to S.

show abstract

Kernels in edge-colored digraphs

Cited by 38 publications

References 3 publications

Kernels in the closure of coloured digraphs

Kernels in the closure of coloured digraphs

Kernels by properly colored paths in arc-colored digraphs

King-Serf Duo by Monochromatic Paths in $k$-Edge-Coloured Tournaments

Contact Info

Product

Resources

About