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

• Volume: 29, Issue: 2, page 337-347
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Hortensia Galeana-Sánchez, R. Rojas-Monroy, and B. Zavala. "Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs." Discussiones Mathematicae Graph Theory 29.2 (2009): 337-347. <http://eudml.org/doc/270653>.

@article{HortensiaGaleana2009,
abstract = {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.},
author = {Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {m-coloured digraph; 3-quasitransitive digraph; kernel by monochromatic paths; γ-cycle; quasi-monochromatic digraph; -coloured digraph; -cycle},
language = {eng},
number = {2},
pages = {337-347},
title = {Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs},
url = {http://eudml.org/doc/270653},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Hortensia Galeana-Sánchez
AU - R. Rojas-Monroy
AU - B. Zavala
TI - Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs
JO - Discussiones Mathematicae Graph Theory
PY - 2009
VL - 29
IS - 2
SP - 337
EP - 347
AB - 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.
LA - eng
KW - m-coloured digraph; 3-quasitransitive digraph; kernel by monochromatic paths; γ-cycle; quasi-monochromatic digraph; -coloured digraph; -cycle
UR - http://eudml.org/doc/270653
ER -

## References

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1. [1] J. Bang-Jensen and J. Huang, Quasi-transitive digraphs, J. Graph Theory 20 (1995) 141-161, doi: 10.1002/jgt.3190200205. Zbl0832.05048
2. [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. Zbl0955.05048
3. [3] C. Berge, Graphs (North Holland, Amsterdam, New York, 1985).
4. [4] P. Duchet, Graphes noyau-parfaits, Ann. Discrete Math. 9 (1980) 93-101, doi: 10.1016/S0167-5060(08)70041-4.
5. [5] P. Duchet, Classical Perfect Graphs, An introduction with emphasis on triangulated and interval graphs, Ann. Discrete Math. 21 (1984) 67-96. Zbl0558.05038
6. [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. Zbl0456.05032
7. [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. Zbl0857.05054
8. [8] H. Galeana-Sánchez, Kernels in edge coloured digraphs, Discrete Math. 184 (1998) 87-99, doi: 10.1016/S0012-365X(97)00162-3. Zbl0958.05061
9. [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. Zbl0529.05024
10. [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. Zbl0593.05034
11. [11] H. Galeana-Sánchez, R. Rojas-Monroy and B. Zavala, Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs, submitted. Zbl1217.05089
12. [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. Zbl0105.35503
13. [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. Zbl0654.05033
14. [14] M. Richardson, Solutions of irreflexive relations, Ann. Math. 58 (1953) 573, doi: 10.2307/1969755. Zbl0053.02902
15. [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. Zbl0053.02903
16. [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. Zbl0488.05036

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