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

Hortensia Galeana-Sanchez; Rocío Rojas-Monroy

Discussiones Mathematicae Graph Theory (2008)

- Volume: 28, Issue: 2, page 285-306
- ISSN: 2083-5892

## Access Full Article

top## Abstract

top## How to cite

topHortensia Galeana-Sanchez, and Rocío Rojas-Monroy. "Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments." Discussiones Mathematicae Graph Theory 28.2 (2008): 285-306. <http://eudml.org/doc/270359>.

@article{HortensiaGaleana2008,

abstract = {
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.
},

author = {Hortensia Galeana-Sanchez, Rocío Rojas-Monroy},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {kernel; kernel by monochromatic paths; bipartite tournament},

language = {eng},

number = {2},

pages = {285-306},

title = {Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments},

url = {http://eudml.org/doc/270359},

volume = {28},

year = {2008},

}

TY - JOUR

AU - Hortensia Galeana-Sanchez

AU - Rocío Rojas-Monroy

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

JO - Discussiones Mathematicae Graph Theory

PY - 2008

VL - 28

IS - 2

SP - 285

EP - 306

AB -
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.

LA - eng

KW - kernel; kernel by monochromatic paths; bipartite tournament

UR - http://eudml.org/doc/270359

ER -

## References

top- [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. Zbl0456.05032
- [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. Zbl0529.05024
- [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. Zbl0958.05061
- [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. Zbl0990.05059
- [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. Zbl1049.05042
- [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. Zbl1042.05039
- [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. Zbl1042.05049
- [11] M. Richardson, Solutions of irreflexive relations, Ann. Math. 58 (1953) 573, doi: 10.2307/1969755. Zbl0053.02902
- [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. Zbl0654.05033
- [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. Zbl0488.05036
- [14] J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, Princeton, 1944). Zbl0063.05930

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.