Geodesic
γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 3, pp. 493-507.

Voir la notice de l'article provenant de la source Library of Science

We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = (u0, u1, . . ., un), such that ui ≠ uj if i ≠ j and for every i ∈ 0, 1, . . ., n there is a uiui+1-monochromatic path in D and there is no ui+1ui-monochromatic path in D (the indices of the vertices will be taken mod n+1). 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 path between them and; (ii) for every vertex x ∈ V (D) N there is a vertex y ∈ N such that there is an xy-monochromatic path. Let D be a finite m-coloured digraph. Suppose that C1,C2 is a partition of C, the set of colours of D, and Di will be the spanning subdigraph of D such that A(Di) = a ∈ A(D) | colour(a) ∈ Ci. In this paper, we give some sufficient conditions for the existence of a kernel by monochromatic paths in a digraph with the structure mentioned above. In particular we obtain an extension of the original result by B. Sands, N. Sauer and R. Woodrow that asserts: Every 2-coloured digraph has a kernel by monochromatic paths. Also, we extend other results obtained before where it is proved that under some conditions an m-coloured digraph has no γ-cycles.
Keywords: digraph, kernel, kernel by monochromatic paths, γ-cycle 
@article{DMGT_2013_33_3_a1,
     author = {Casas-Bautista, Enrique and Galeana-S\'anchez, Hortensia and Rojas-Monroy, Roc{\'\i}o},
     title = {\ensuremath{\gamma}-Cycles {And} {Transitivity} {By} {Monochromatic} {Paths} {In} {Arc-Coloured} {Digraphs}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {493--507},
     publisher = {mathdoc},
     volume = {33},
     number = {3},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2013_33_3_a1/}
}
TY  - JOUR
AU  - Casas-Bautista, Enrique
AU  - Galeana-Sánchez, Hortensia
AU  - Rojas-Monroy, Rocío
TI  - γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2013
SP  - 493
EP  - 507
VL  - 33
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2013_33_3_a1/
LA  - en
ID  - DMGT_2013_33_3_a1
ER  - 
%0 Journal Article
%A Casas-Bautista, Enrique
%A Galeana-Sánchez, Hortensia
%A Rojas-Monroy, Rocío
%T γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs
%J Discussiones Mathematicae. Graph Theory
%D 2013
%P 493-507
%V 33
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2013_33_3_a1/
%G en
%F DMGT_2013_33_3_a1
Casas-Bautista, Enrique; Galeana-Sánchez, Hortensia; Rojas-Monroy, Rocío. γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 3, pp. 493-507. http://geodesic.mathdoc.fr/item/DMGT_2013_33_3_a1/

[1] J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications (Springer, London, 2001).

[2] C. Berge, Graphs (North-Holland, Amsterdam, 1985).

[3] 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

[4] P. Delgado-Escalante and H. Galena-Sánchez, Kernels and cycles’ subdivisions in arc-colored tournaments, Discuss. Math. Graph Theory 29 (2009) 101-117. doi:10.7151/dmgt.1435

[5] P. Delgado-Escalante and H. Galena-Sánchez, On monochromatic paths and bicolored subdigraphs in arc-colored tournaments, Discuss. Math. Graph Theory 31 (2011) 791-820. doi:10.7151/dmgt.1580

[6] P. Duchet, Graphes noyau - parfaits, Ann. Discrete Math. 9 (1980) 93-101. doi:10.1016/S0167-5060(08)70041-4

[7] P. Duchet, Classical perfect graphs, An introduction with emphasis on triangulated and interval graphs, Ann. Discrete Math. 21 (1984) 67-96.

[8] P. Duchet and H. Meynel, A note on kernel-critical graphs, Discrete Math. 33 (1981) 103-105. doi:10.1016/0012-365X(81)90264-8

[9] H. Galena-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

[10] H. Galena-Sánchez, Kernels in edge-coloured digraphs, Discrete Math. 184 (1998) 87-99. doi:10.1016/S0012-365X(97)00162-3

[11] H. Galena-Sánchez and J.J. García-Ruvalcaba, Kernels in the closure of coloured digraphs, Discuss. Math. Graph Theory 20 (2000) 243-254. doi:10.7151/dmgt.1123

[12] H. Galeana-Sánchez, J.J. García-Ruvalcaba, On graphs all of whose {C3, T3}-free arc colorations are kernel perfect, Discuss. Math. Graph Theory 21 (2001) 77-93. doi:10.7151/dmgt.1134

[13] H. Galena-Sánchez, G. Gaytán-Gómez and R. Rojas-Monroy, Monochromatic cycles and monochromatic paths in arc-coloured digraphs, Discuss. Math. Graph Theory 31 (2011) 283-292. doi:10.7151/dmgt.1545

[14] H. Galena-Sánchez, V. Neumann-Lara, On kernels and semikernels of digraphs, Discrete Math. 48 (1984) 67-76. doi:10.1016/0012-365X(84)90131-6

[15] H. Galeana-Sánchez, V. Neumann-Lara, On kernel-perfect critical digraphs, Discrete Math. 59 (1986) 257-265. doi:10.1016/0012-365X(86)90172-X

[16] H. Galeana-Sánchez and R. Rojas-Monroy, A counterexample to a conjecture on edge-coloured tournaments, Discrete Math. 282 (2004) 275-276. doi:10.1016/j.disc.2003.11.015

[17] H. Galeana-Sánchez and R. Rojas-Monroy, On monochromatic paths and monochromatic 4-cycles in edge coloured bipartite tournaments, Discrete Math. 285 (2004) 313-318. doi:10.1016/j.disc.2004.03.005

[18] H. Galeana-Sánchez, R. Rojas-Monroy, Independent domination by monochromatic paths in arc coloured bipartite tournaments, AKCE J. Graphs. Combin. 6 (2009) 267-285.

[19] H. Galeana-Sánchez and R. Rojas-Monroy, Monochromatic paths and monochromatic cycles in edge-coloured k-partite tournaments, Ars Combin. 97A (2010) 351-365.

[20] H. Galena-Sánchez, R. Rojas-Monroy and B. Zavala, Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs, Discuss. Math. Graph Theory 29 (2009) 337-347. doi:10.7151/dmgt.1450

[21] H. Galena-Sánchez, R. Rojas-Monroy and B. Zavala, Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs, Discuss. Math. Graph Theory 30 (2010) 545-553. doi:10.7151/dmgt.1512

[22] G. Hahn, P. Ille and R. Woodrow, Absorbing sets in arc-coloured tournaments, Discrete Math. 283 (2004) 93-99. doi:10.1016/j.disc.2003.10.024

[23] J.M. Le Bars, Counterexample of the 0 − 1 law for fragments of existential secondorder logic; an overview, Bull. Symbolic Logic 6 (2000) 67-82. doi:10.2307/421076

[24] J.M. Le Bars, The 0−1 law fails for frame satisfiability of propositional model logic, in: Proceedings of the 17th Symposium on Logic in Computer Science (2002) 225-234. doi:10.1109/LICS.2002.1029831

[25] J. von Leeuwen, Having a Grundy numbering is NP-complete, Report 207 Computer Science Department, Pennsylvania State University, University Park, PA (1976).

[26] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, Princeton, 1944).

[27] 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

[28] 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

[29] I. Włoch, On kernels by monochromatic paths in the corona of digraphs, Cent. Eur. J. Math. 6 (2008) 537-542. doi:10.7151/s11533-008-0044-6

[30] I. Włoch, On imp-sets and kernels by monochromatic paths in duplication, Ars Combin. 83 (2007) 93-99.