# Kernels in the closure of coloured digraphs

Hortensia Galeana-Sánchez; José de Jesús García-Ruvalcaba

Discussiones Mathematicae Graph Theory (2000)

- Volume: 20, Issue: 2, page 243-254
- ISSN: 2083-5892

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topHortensia Galeana-Sánchez, and José de Jesús García-Ruvalcaba. "Kernels in the closure of coloured digraphs." Discussiones Mathematicae Graph Theory 20.2 (2000): 243-254. <http://eudml.org/doc/270579>.

@article{HortensiaGaleana2000,

abstract = {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 T₃ and C₃ 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 T₃ or C₃, then ξ(D) is a kernel-perfect digraph.},

author = {Hortensia Galeana-Sánchez, José de Jesús García-Ruvalcaba},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {kernel; closure; tournament; -coloured digraph},

language = {eng},

number = {2},

pages = {243-254},

title = {Kernels in the closure of coloured digraphs},

url = {http://eudml.org/doc/270579},

volume = {20},

year = {2000},

}

TY - JOUR

AU - Hortensia Galeana-Sánchez

AU - José de Jesús García-Ruvalcaba

TI - Kernels in the closure of coloured digraphs

JO - Discussiones Mathematicae Graph Theory

PY - 2000

VL - 20

IS - 2

SP - 243

EP - 254

AB - 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 T₃ and C₃ 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 T₃ or C₃, then ξ(D) is a kernel-perfect digraph.

LA - eng

KW - kernel; closure; tournament; -coloured digraph

UR - http://eudml.org/doc/270579

ER -

## References

top- [1] 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. Zbl0721.05027
- [2] H. Galeana-Sánchez, On monochromatic paths and monochromatic cycles in edge coloured tournaments, Discrete Math. 156 (1996) 103-112, doi: 10.1016/0012-365X(95)00036-V. Zbl0857.05054
- [3] H. Galeana-Sánchez, Kernels in edge coloured digraphs, Discrete Math. 184 (1998) 87-99, doi: 10.1016/S0012-365X(97)00162-3. Zbl0958.05061
- [4] H. Galeana-Sánchez and J.J. García-Ruvalcaba, On graph all of whose {C₃, T₃}-free arc colorations are kernel-perfect, submitted. Zbl0990.05060
- [5] Shen Minggang, On monochromatic paths in m-coloured tournaments, J. Combin. Theory (B) 45 (1988) 108-111, doi: 10.1016/0095-8956(88)90059-7. Zbl0654.05033
- [6] B. Sands, N. Sauer and R. Woodrow, On monochromatic paths in edge-coloured digraphs, J. Combin. Theory (B) 33 (1982) 271-275, doi: 10.1016/0095-8956(82)90047-8. Zbl0488.05036

## Citations in EuDML Documents

top- Hortensia Galeana-Sanchez, Rocío Rojas-Monroy, Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments
- Hortensia Galeana-Sánchez, Guadalupe Gaytán-Gómez, Rocío Rojas-Monroy, Monochromatic cycles and monochromatic paths in arc-colored digraphs
- Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala, Monochromatic paths and monochromatic sets of arcs in bipartite tournaments
- Hortensia Galeana-Sánchez, Laura Pastrana Ramírez, Hugo Alberto Rincón Mejía, Kernels in monochromatic path digraphs
- Hortensia Galeana-Sánchez, Guadalupe Gaytán-Gómez, Rocío Rojas-Monroy, γ-Cycles In Arc-Colored Digraphs
- Enrique Casas-Bautista, Hortensia Galeana-Sánchez, Rocío Rojas-Monroy, γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs

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