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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1123,
author = {Hortensia Galeana-S\'anchez and Jos\'e de Jes\'us Garc\'\i a-Ruvalcaba},
title = {Kernels in the closure of coloured digraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {20},
year = {2000},
pages = {243-254},
zbl = {0990.05059},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1123}
}
Hortensia Galeana-Sánchez; José de Jesús García-Ruvalcaba. Kernels in the closure of coloured digraphs. Discussiones Mathematicae Graph Theory, Tome 20 (2000) pp. 243-254. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1123/
[000] [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. | Zbl 0721.05027
[001] [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. | Zbl 0857.05054
[002] [3] H. Galeana-Sánchez, Kernels in edge coloured digraphs, Discrete Math. 184 (1998) 87-99, doi: 10.1016/S0012-365X(97)00162-3. | Zbl 0958.05061
[003] [4] H. Galeana-Sánchez and J.J. García-Ruvalcaba, On graph all of whose {C₃, T₃}-free arc colorations are kernel-perfect, submitted. | Zbl 0990.05060
[004] [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. | Zbl 0654.05033
[005] [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. | Zbl 0488.05036