Grundy Chromatic Number of Shuriken Graphs and Applications to Regulatory Network Modeling - M. Kamalnath; T, Muthukani Vairavel (Statistics, Optimization & Information Computing, 2026)
The concept of Grundy coloring plays an important role in the study of graph coloring and its applications in sequential allocation problems. In this paper, we study the Grundy (first-fit) chromatic number of the Shuriken graph Sh_n and some of its associated derived graphs obtained through standard graph operations. In particular, we determine the Grundy chromatic numbers of the line graph L(Sh_n), the middle graph M(Sh_n), and the total graph T(Sh_n). The results are obtained by utilizing the structural properties of the Shuriken graph together with the principles of Grundy coloring and suitable upper bound techniques. In addition, we discuss the relevance of these graphs in the framework of Gene Regulatory Networks (GRNs), where the Grundy chromatic number may represent the maximum level of hierarchical activations arising in sequential regulatory interactions. Thus, the Shuriken graph and its related derived graphs provide meaningful mathematical models for studying interaction complexity and stability in biological systems.