On BMRN*-colouring of planar digraphs

International audience 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.

A Dynamic TDMA Scheduling Strategy for MANETs Based on Service Priority

Physical resource allocation strategy is a key factor affecting the performance of a mobile ad hoc network (MANET), which serves as a network model widely used in the sensor and detection field. For various perceived service information, it is significant for the allocation strategy to adapt to the dynamic service requirements and prioritize resource access for the service information with high priority to guarantee its real-time performance. In this paper, a novel dynamic time division multiple access (TDMA) scheduling strategy is proposed for MANETs. Firstly, a service priority-based dynamic TDMA scheduling (SP-DS) algorithm is presented, which introduces the service priority as a reference factor for slot allocation and fully considers the transmission throughput and end-to-end delay performance. Moreover, for improving the slot use of the whole system, a modified distributed color constraint heuristic (MD-CCH) scheme is presented to optimize the frame structure. The SP-DS and MD-CCH algorithms are combined to form the novel strategy. Simulation results have demonstrated that the proposed strategy has better performance in the slot use, slot allocation efficiency, end-to-end delay and transmission throughput compared with the existing slot allocation algorithms.