Research Article Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach A. El-Mesady 1 and Omar Bazighifan 2,3 1 Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt 2 Department of Mathematics, Faculty of Science, Hadhramaut University, Hadhramaut, Al Mukalla 50512, Yemen 3 Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout 50512, Yemen Correspondence should be addressed to Omar Bazighifan; o.bazighifan@gmail.com Received 27 May 2022; Accepted 24 June 2022; Published 18 July 2022 Academic Editor: Miaochao Chen Copyright © 2022 A. El-Mesady and Omar Bazighifan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For applied scientists and engineers, graph theory is a strong and vital tool for evaluating and inventing solutions for a variety of issues. Graph theory is extremely important in complex systems, particularly in computer science. Many scientific areas use graph theory, including biological sciences, engineering, coding, and operational research. A strategy for the orthogonal labelling of a bipartite graph G with n edges has been proposed in the literature, yielding cyclic decompositions of balanced complete bipartite graphs K n,n by the graph G. A generalization to circulant-balanced complete multipartite graphs K n,n,⋯,n |fflfflfflffl{zfflfflfflffl} m ; m, n ≥ 2, is our objective here. In this paper, we expand the orthogonal labelling approach used to generate cyclic decompositions for K n,n to a generalized orthogonal labelling approach that may be used for decomposing K n,n,⋯,n |fflfflffl{zfflfflfflffl} m . We can decompose K n,n,⋯,n |fflfflfflffl{zfflfflfflffl} m into distinct graph classes based on the proposed generalized orthogonal labelling approach. 1. Introduction As is well known, discrete mathematics is a field of mathemat- ics that deals with countable processes and components. One of the most significant and intriguing disciplines in discrete math- ematics is graph theory [1–3]. Graph theory is the study of structural models called graphs, which are made up of a collec- tion of vertices and edges. Graph theory is extremely important in complex systems, particularly in computer science. Many scientific areas use graph theory, including engineering, coding [4, 5], operational research, biological sciences, and manage- ment sciences. For applied scientists and engineers, graph the- ory is a strong and vital science for evaluating and inventing solutions for a variety of issues. Graphs have recently been uti- lized as structural models for characterizing World Wide Web connections and the number of links necessary to move between web pages [6]. Circulant graphs are a significant category of graphs [7–10]. Circulant graphs have gained a lot of attention in recent decades. The circulant graphs class includes complete graphs and classic rings topologies. The algebraic properties of circulant graphs have been studied in thousands of publications. Circu- lant graphs have been handled in a variety of graph applications, including wide area communication graphs, local area com- puter graphs, parallel processing architectures, very large-scale integrated circuit design, and distributed computing [11–13]. Several traditional parallel and distributed systems were built on the foundation of circulant graphs [14–16]. Circulant graphs have a wide range of practical uses, such as a structure in chemical reaction models [17], multiprocessor cluster Hindawi Journal of Function Spaces Volume 2022, Article ID 2017936, 17 pages https://doi.org/10.1155/2022/2017936