## 1 Introduction  
Due to its vast functions, the global system of the mobile communication (GSM) cellular network has experienced a rapid increase in the number of subscribers. As the number of users increases, the need for efficient and effective planning of the GSM's \"limited\" frequency spectrum is inevitable, particularly in densely populated areas. As such, there are ongoing discussions regarding frequency (channel) allocation methods to resolve the challenges of channel allocation, which presents itself as an NP complete problem. Discourse on channel allocation has gained momentum in recent years, so that demand for a ratio spectrum (which is and will always remain a scarce commodity) has intensified [1; 2; 3]. Although the world has adopted faster communication technology, such as the 5G network, the structure of the GSM is the foundation upon which most new network structures are built, especially for voice and text messaging services [4; 5]. While several advanced radio technologies are now popular, cellular network providers in some developing countries still adopt the Manhattan distance technique for GSM. The main reason for this is the high cost of new technologies such as 4G and 5G, which are yet be fully adopted. Nevertheless, since the range of cellular communication is of the utmost importance, newer methods have been developed to further increase the range of cellular networks. It is believed that the greater the level of optimization, the better it is for network communication and development in terms of cell-by-cell connectivity. In a study [6], a geometrical technique for determining cell range was developed. The technique utilizes a so-called Voronoi tessellation. The specific inputs needed for this technique include the location of the site, the beam width of the horizontal antennae, and the antenna azimuths. While this method has been shown to improve on the conventional technique, this improvement is still largely technically tied to the GSM basics. In GSM technology, the allocation or assignment of planning frequency is crucial in the modern telecommunication regime. This is useful not only at the system commencement phase, but also at later stages of network modification or expansion, which should cater for high levels of interference and mass usage, among other factors [4; 5; 6]. Three channel allocation schemes have been identified in the literature [7; 8], namely dynamic, fixed, and hybrid. Nevertheless, assigning a frequency remains a challenge [9]. Assume the existence of a set of channels which aids frequency transmission to a group of transmitters. In the vicinity of a dual transmitter, individual transmitters must use a different channel so as to maintain the needed transmission quality. The lowest uniqueness in frequencies assigned to dual transmitters in proximity is a function of their level of interference. The goal of frequency assignment is to locate the best possible assignment (given a frequency that is quite small) without compromising the quality of communication. The channel allocation problem is a class issue which is NP complete, implying that no existing algorithms (polynomial time) can solve this problem [10; 11; 12; 13]  
**Definition 1**: A graph is termed complete if it is simply constructed and undirected and has a unique pair of vertices linked by a different edge. In this study, \(C_{n}\) denotes a hypothetical scenario of a complete graph with \(n\) vertices, where \(a,b\in T\), and \((a,b)\in H\) represent the vertices [14]. **Definition 2**: Let \(U\) denote a graph that is undirected, let \(T\) represent a group of nodes, let \(E\) represent a group of lines connecting the nodes, and let \(C\) represent a group of colors. The process of mapping \(CT\) so that no corresponding vertices within the undirected graph, \(U\), are allotted the same channel is referred to as \"graph coloring\".