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Discussiones Mathematicae Graph Theory 25(3) (2005) 407-417
DOI: https://doi.org/10.7151/dmgt.1292

KERNELS IN MONOCHROMATIC PATH DIGRAPHS

Hortensia Galeana-Sánchez Instituto de Matemáticas, UNAM Universidad Nacional Autónoma de México Ciudad Universitaria 04510, México, D.F. MÉXICO e-mail: hgaleana@matem.unam.mx

Laura Pastrana Ramírez and Hugo Alberto Rincón Mejía Departamento de Matemáticas Facultad de Ciencias Universidad Nacional Autónoma de México Ciudad Universitaria 04510, México, D.F. MÉXICO

Abstract

We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike.

Let D be an m-coloured digraph. A set N ⊆ V(D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions:

(i) for every pair of different vertices u,v ∈ N there is no monochromatic directed path between them and

(ii) for each vertex x ∈ (V(D)−N) there is a vertex y ∈ N such that there is an xy-monochromatic directed path.

In this paper is defined the monochromatic path digraph of D, MP(D), and the inner m-colouration of MP(D). Also it is proved that if D is an m-coloured digraph without monochromatic directed cycles, then the number of kernels by monochromatic paths in D is equal to the number of kernels by monochromatic paths in the inner m-colouration of MP(D). A previous result is generalized.

Keywords: kernel, line digraph, kernel by monochromatic paths, monochromatic path digraph, edge-coloured digraph.

2000 Mathematic Subject Classification: 05C20.

References

[1]	C. Berge, Graphs (North-Holland, Amsterdam, 1985).
[2]	C. Berge and A. Ramachandra Rao, A combinatorial problem in logic, Discrete Math. 17 (1977) 23-26, doi: 10.1016/0012-365X(77)90018-8.
[3]	P. Duchet, A sufficient condition for a digraph to be kernel-perfect, J. Graph Theory 11 (1987) 81-85, doi: 10.1002/jgt.3190110112.
[4]	P. Duchet and H. Meyniel, Une généralization du théoréme de Richardson sur l'existence du noyaux dans les graphes orientés, Discrete Math. 43 (1983) 21-27, doi: 10.1016/0012-365X(83)90017-1.
[5]	H. Galeana-Sánchez and V. Neumann-Lara, On kernels and semikernels of digraphs, Discrete Math. 48 (1984) 67-76, doi: 10.1016/0012-365X(84)90131-6.
[6]	H. Galeana-Sánchez, L. Pastrana Ramírez and H.A. Rincón-Mejía, Semikernels, quasikernels and Grundy functions in the line digraph, SIAM J. Disc. Math. 1 (1999) 80-83.
[7]	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.
[8]	H. Galeana-Sánchez and Xueliang Li, Semikernels and (k,l)-kernels in digraphs, SIAM J. Discrete Math. 11 (1998) 340-346, doi: 10.1137/S0895480195291370.
[9]	H. Galeana-Sánchez, Kernels in edge coloured digraphs, Discrete Math. 184 (1998) 87-99, doi: 10.1016/S0012-365X(97)00162-3.
[10]	H. Galeana-Sánchez and L. Pastrana Ramírez, Kernels in edge coloured line digraph, Discuss. Math. Graph Theory 18 (1998) 91-98, doi: 10.7151/dmgt.1066.
[11]	H. Galeana-Sánchez and José de Jesús García-Ruvalcaba, Kernels in the closure of coloured digraphs, Discuss. Math. Graph Theory 20 (2000) 243-254, doi: 10.7151/dmgt.1123.
[12]	M. Harminc, Solutions and kernels of a directed graph, Math. Slovaca 32 (1982) 263-267.
[13]	J. Von Neumann and O. Morgenstern, Theory of games and economic behavior (Princeton University Press, Princeton, NJ, 1944).
[14]	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.
[15]	S. Minggang, On monochromatic paths in m-coloured tournaments, J. Combin. Theory (B) 45 (1988) 108-111, doi: 10.1016/0095-8956(88)90059-7.

Received 3 August 2004
Revised 10 December 2004