Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

Abstract

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 $K_{\underset{m}{\underbrace{n,n,\cdots ,n}}};m,n\ge 2,$ 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 $K_{\underset{m}{\underbrace{n,n,\cdots ,n}}}$ . We can decompose $K_{\underset{m}{\underbrace{n,n,\cdots ,n}}}$ into distinct graph classes based on the proposed generalized orthogonal labelling approach.