Acyclic and star coloring parameters of fractal cubic networks

Interconnection networks are more vital in telecommunications because of the significant raise in the demand for high-speed networks as a result of the widespread use of computers and the growth of the internet. The hypercube is a versatile network with outstanding qualities that are important for developing extensively in parallel and distributed systems, which include smaller size diameter, recursive structure, symmetry, regularity, low degree, and scalability. In the realm of distributed systems, scalability is seen as an elasticity component of interconnection networks. Fractal cubic networks, a new and novel variant of hypercubes, were recently investigated and have very important qualities such as scalability and better bisection width than traditional hypercubes. The task of assigning channels can be represented as a graph coloring problem. The vertices of a graph represent the transmitters, and if two transmitters are in close proximity to each other, their corresponding vertices are considered nearby. Wavelength assignment enhances the efficiency of wavelength-routed networks by determining routes and assigning wavelengths to connection requests while adhering to network topology and wavelength constraints. Investigation of the acyclic, acyclic edge, star, and star edge chromatic numbers for this newly proposed interconnection network, which is in striking contrast to the situation with hypercubes, where these invariants are intrinsically difficult. In this paper, we establish that for Fractal Cubic Networks (FCNs), the acyclic chromatic number is [Formula: see text] for [Formula: see text]. Additionally, for [Formula: see text], the star chromatic number [Formula: see text] and the acyclic edge chromatic number [Formula: see text] are both 4, while the star edge chromatic number is [Formula: see text].