# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2019: 0.755

SCImago Journal Rank (SJR) 2019: 0.600

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# Discussiones Mathematicae Graph Theory

## 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

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