A note on vertex partitions

Discrete Applied Mathematics, 2013

2002

International Journal of Mathematics and Mathematical Sciences, 1994

Combinatorica, 2007

Lecture Notes in Computer Science, 1999

Information Processing Letters, 1988

Journal of Graph Theory, 2015

Combinatorics, Probability and Computing, 1998

We consider the function χ(Gk), defined to be the smallest number of colours that can colour a graph G in such a way that no vertices of distance at most k receive the same colour. In particular we shall look at how small a value this function can take in terms of the order and diameter of G. We get general bounds for this and tight bounds for the cases k=2 and k=3.

ResearchGate, 2024

This paper aims to present a new prototype designed to reduce the noise generated by drone or UAV propellers. This prototype is called the Drones Anti-Noise Propellers Compartments (DANP-Compartments). The DANP-Compartments can reduce the disturbing noise of propellers, improve the drone's flight performance, prevent injuries in congested areas, and protect pilots. This device introduces new features and an alternative system to reduce the loud noise produced by drone propellers in both open and congested places and to enhance performance in critical weather conditions. In our experiment, the DANP-Compartments were used on hydrones (drones designed for water). The main objective was to evaluate how the DANP-Compartments could significantly reduce the noise of the hydrones and protect the motors in rainy and snowy conditions. This preliminary experimental prototype offers potential advantages for both civilian and military drone industries. Additionally, we propose a mathematical and graphical model to evaluate propeller noise under chaotic noise conditions using nonlinear irregular strokes coordinate plane. This new coordinate plane can visualize and analyse noise from a multidimensional perspective. We believe that the chaotic noise nonlinear irregular strokes coordinate plane can play an important role in the theoretical and analytical study of anti-noise drone devices during takeoff , cruising, and landing.