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## Discussiones Mathematicae Graph Theory

2008 | 28 | 2 | 285-306
Tytuł artykułu

### Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A directed cycle is called quasi-monochromatic if with at most one exception all of its arcs are coloured alike. 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 every vertex x ∈ V(D)∖N there is a vertex y ∈ N such that there is an xy-monochromatic directed path.
In this paper it is proved that if D is an m-coloured bipartite tournament such that: every directed cycle of length 4 is quasi-monochromatic, every directed cycle of length 6 is monochromatic, and D has no induced particular 6-element bipartite tournament T̃₆, then D has a kernel by monochromatic paths.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
285-306
Opis fizyczny
Daty
wydano
2008
otrzymano
2007-07-06
poprawiono
2008-04-10
zaakceptowano
2008-04-10
Twórcy
• Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México, D.F. 04510, México
autor
• Facultad de Ciencias, Universidad Autónoma del Estado de México, Instituto Literario No. 100, Centro, 50000, Toluca, Edo. de México, México
Bibliografia
• [1] C. Berge, Graphs (North-Holland, Amsterdam, 1985).
• [2] P. Duchet, Graphes Noyau-Parfaits, Ann. Discrete Math. 9 (1980) 93-101, doi: 10.1016/S0167-5060(08)70041-4.
• [3] P. Duchet and H. Meyniel, A note on kernel-critical graphs, Discrete Math. 33 (1981) 103-105, doi: 10.1016/0012-365X(81)90264-8.
• [4] 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.
• [5] H. Galeana-Sánchez, On monochromatic paths and monochromatics cycles in edge coloured tournaments, Discrete Math. 156 (1996) 103-112, doi: 10.1016/0012-365X(95)00036-V.
• [6] H. Galeana-Sánchez, Kernels in edge-coloured digraphs, Discrete Math. 184 (1998) 87-99, doi: 10.1016/S0012-365X(97)00162-3.
• [7] H. Galeana-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.
• [8] 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.
• [9] 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.
• [10] 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.
• [11] M. Richardson, Solutions of irreflexive relations, Ann. Math. 58 (1953) 573, doi: 10.2307/1969755.
• [12] 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.
• [13] 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.
• [14] J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, Princeton, 1944).
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