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

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 (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.

How to cite

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Hortensia 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

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  10. [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
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