On Domatic Number of Some Rotationally Symmetric Graphs

Abstract

Domination is a well-known graph theoretic concept due to its significant real-world applications in several domains, such as design and communication network analysis, coding theory, and optimization. For a connected graph $\mathrm{\Gamma }=\left(V,E\right)$ , a subset $U$ of $V\left(\mathrm{\Gamma }\right)$ is called a dominating set if every member present in $V-U$ is adjacent to at least one member in $U$ . The domatic partition is the partition of the vertices $V\left(\mathrm{\Gamma }\right)$ into the disjoint dominating set. The domatic number of the graph $\mathrm{\Gamma }$ is the maximum cardinality of the disjoint dominating sets. In this paper, we improved the results for the middle and central graphs of a cycle, respectively. Furthermore, we discuss the domatic number for some other cycle-related graphs and graphs of convex polytopes.