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.
Highlights
The colouring concepts studied in this work are motivated by the following real-life problem
Towards Question 1.2, we prove that deciding whether BMRN∗(D, B) ≤ k holds for a given planar backboned digraph (D, B) is NP-hard for every k ∈ {4, . . . , 6}, even when restricted to planar spanned digraphs
Definitions, notation and terminology An undirected graph H is a minor of another undirected graph G if H can be obtained from G by deleting edges, deleting vertices, and contracting edges
Summary
The colouring concepts studied in this work are motivated by the following real-life problem. They considered algorithmic aspects related to the problem of determining the BMRN-index or the BMRN∗-index of a given spanned digraph, which in general is NP-hard, even if one is allowed to construct the backbone from a given root They gave a number of more specific results for particular classes of digraphs, such as bounded-degree digraphs, outerplanar digraphs, and more generally planar digraphs. As a first result in this paper, we answer negatively to Question 1.1 by exhibiting, in Section 2, a planar spanned digraph (D, T ) verifying BMRN(D, T ) = BMRN∗(D, T ) = 8 This shows that the upper bound above is tight. It seems judicious to investigate the behaviour of the BMRN∗-index of planar backboned digraphs when small cycles are excluded, which is a classical aspect in graph colouring theory. Whenever referring to a digraph notion or notation for a backboned digraph (D, B), we implicitly refer to the corresponding notion or notation for D
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More From: Discrete Mathematics & Theoretical Computer Science
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