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## Discussiones Mathematicae Graph Theory

2009 | 29 | 2 | 337-347
Tytuł artykułu

### Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs

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 is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v ∉ N there is a monochromatic path from v to N. We denote by A⁺(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured 3-quasitransitive digraph such that for every vertex u of D, A⁺(u) is monochromatic and D satisfies some colouring conditions over one subdigraph of D of order 3 and two subdigraphs of D of order 4, then D has a kernel by monochromatic paths.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
337-347
Opis fizyczny
Daty
wydano
2009
otrzymano
2007-11-06
poprawiono
2009-02-26
zaakceptowano
2009-02-27
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, Centro 50000, Toluca, Edo. de México, México
autor
• Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México, D.F. 04510, México
Bibliografia
• [1] J. Bang-Jensen and J. Huang, Quasi-transitive digraphs, J. Graph Theory 20 (1995) 141-161, doi: 10.1002/jgt.3190200205.
• [2] J. Bang-Jensen and J. Huang, Kings in quasi-transitive digraphs, Discrete Math. 185 (1998) 19-27, doi: 10.1016/S0012-365X(97)00179-9.
• [3] C. Berge, Graphs (North Holland, Amsterdam, New York, 1985).
• [4] P. Duchet, Graphes noyau-parfaits, Ann. Discrete Math. 9 (1980) 93-101, doi: 10.1016/S0167-5060(08)70041-4.
• [5] P. Duchet, Classical Perfect Graphs, An introduction with emphasis on triangulated and interval graphs, Ann. Discrete Math. 21 (1984) 67-96.
• [6] 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.
• [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, Kernels in edge coloured digraphs, Discrete Math. 184 (1998) 87-99, doi: 10.1016/S0012-365X(97)00162-3.
• [9] H. Galena-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.
• [10] H. Galeana-Sánchez and V. Neumann-Lara, On kernel-perfect critical digraphs, Discrete Math. 59 (1986) 257-265, doi: 10.1016/0012-365X(86)90172-X.
• [11] H. Galeana-Sánchez, R. Rojas-Monroy and B. Zavala, Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs, submitted.
• [12] Ghouilá-Houri, Caractérisation des graphes non orientés dont on peut orienter les aretes de maniére à obtenir le graphe d'une relation d'ordre, C.R. Acad. Sci. Paris 254 (1962) 1370-1371.
• [13] 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.
• [14] M. Richardson, Solutions of irreflexive relations, Ann. Math. 58 (1953) 573, doi: 10.2307/1969755.
• [15] M. Richardson, Extensions theorems for solutions of irreflexive relations, Proc. Nat. Acad. Sci. USA 39 (1953) 649, doi: 10.1073/pnas.39.7.649.
• [16] 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.
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