Julien Bensmail ; Foivos Fioravantes - On BMRN*-colouring of planar digraphs

dmtcs:5798 - Discrete Mathematics & Theoretical Computer Science, February 25, 2021, vol. 23 no. 1 - https://doi.org/10.46298/dmtcs.5798
On BMRN*-colouring of planar digraphsArticle

Authors: Julien Bensmail 1; Foivos Fioravantes ORCID1

  • 1 Combinatorics, Optimization and Algorithms for Telecommunications

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.

Volume: vol. 23 no. 1
Section: Graph Theory
Published on: February 25, 2021
Accepted on: January 25, 2021
Submitted on: October 1, 2019
Keywords: BMRN*-colouring,planar digraphs,TDMA scheduling,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]

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