Skip to Main content Skip to Navigation
Journal articles

On BMRN*-colouring of planar digraphs

Julien Bensmail 1 Foivos Fioravantes 1
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In a recent work, Bensmail, Blanc, Cohen, Havet and Rocha, motivated by applications for TDMA scheduling problems, have introduced the notion of BMRN*-colouring of digraphs, which is a type of arc-colouring with particular colouring constraints. In particular, they gave a special focus to planar digraphs. They notably proved that every planar digraph can be 8-BMRN*-coloured, while there exist planar digraphs for which 7 colours are needed in a BMRN*-colouring. They also proved that the problem of deciding whether a planar digraph can be 3-BMRN*-coloured is NP-hard. In this work, we pursue these investigations on planar digraphs, in particular by answering some of the questions left open by the authors in that seminal work. We exhibit planar digraphs needing 8 colours to be BMRN*-coloured, thus showing that the upper bound of Bensmail, Blanc, Cohen, Havet and Rocha cannot be decreased in general. We also generalize their complexity result by showing that the problem of deciding whether a planar digraph can be k-BMRN*-coloured is NP-hard for every k ∈ {3,...,6}. Finally, we investigate the connection between the girth of a planar digraphs and the least number of colours in its BMRN*-colourings.
Document type :
Journal articles
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-02195028
Contributor : Julien Bensmail <>
Submitted on : Monday, February 15, 2021 - 6:57:25 PM
Last modification on : Tuesday, February 23, 2021 - 3:25:36 AM

File

bmrn-planar.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-02195028, version 6

Collections

Citation

Julien Bensmail, Foivos Fioravantes. On BMRN*-colouring of planar digraphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2021, vol. 23 no. 1 (1), pp.#4. ⟨hal-02195028v6⟩

Share

Metrics

Record views

23

Files downloads

8