DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 20(2) (2000) 243-254
DOI: 10.7151/dmgt.1123

KERNELS IN THE CLOSURE OF COLOURED DIGRAPHS

Hortensia Galeana-Sánchez

Instituto de Matemáticas, UNAM
Circuito Exterior
04510  México, D.F.  México

José de Jesús García-Ruvalcaba

Facultad de Ciencias
Universidad Autónma de Baja California
Km. 103 Carretera Tijuana-Ensenada
Apdo Postal # 1880
22830  Ensenada, B.C.  México

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 xV(D)I   there is a vertex yI 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.

Keywords and phrases: kernel, closure, tournament.

2000 Mathematics Subject Classification: 05C20.

References

[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.
[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.
[3] H. Galeana-Sánchez, Kernels in edge coloured digraphs, Discrete Math. 184 (1998) 87-99, doi: 10.1016/S0012-365X(97)00162-3.
[4] H. Galeana-Sánchez and J.J. García-Ruvalcaba, On graph all of whose {C3, T3}-free arc colorations are kernel-perfect, submitted.
[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.
[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.

Received 31 January 2000
Revised 2 October 2000