On the convergence of the graph sequence { C^m(D) }_m=1^∞ for a multipartite tournament D

Given a positive integer m, the m-step competition graph of a digraph D, denoted by C^m(D), has the same vertex set as D and has an edge between vertices u and v if and only if there exists a vertex w such that there exist directed walks of length m from u to w and from v to w, respectively. In this paper, we completely characterize the convergence of {C^m(D)}_m=1^∞ for a multipartite tournament D based on the last nontrivial strong component of D. Furthermore, not only do we determine the limit in the case of convergence, but also in the event of divergence, we specify how C^m(D) changes periodically depending on the value of m. Our results extend the work of Jung et al. [On the limit of the sequence {C^m (D)}_m=1^∞ for a multipartite tournament D. Discrete Appl. Math., 340:1–13, 2023] which addresses the case of the last strong component being nontrivial, thereby completing the convergence analysis of {C^m(D)}_m=1^∞ for a multipartite tournament D. Our results can also be expressed in terms of matrix sequence {A^m(A^T)^m}_m=1^∞ for the adjacency matrix A of D and this part is also covered in the text.