A Study on Variants of Status Unequal Coloring in Graphs and Its Properties

Abstract

Let be a simple connected graph with vertex set and edge set . The status of a vertex is defined as ∑q≠pd(p, q). A subset of is called a status unequal dominating set (stu‐dominating set) of ; for every , there exists p in such that p and q are adjacent and st(p) ≠ st(q). Among the most admired, extensively explored, and extensively discussed topics in the field of graph theory is graph coloring. This article recommends a new research study on the topic of incorporating stu‐dominating set with coloring. Diverse coloring variants can be found in literature, some of which serve as inspiration is chromatic number, dominator coloring, color class domination, and colorful dominating sets. A novel study on the topic of incorporating stu‐dominating set with coloring is suggested in this article. The stu‐chromatic number, stu‐dominator coloring, stu‐color class domination, and colorful stu‐domination have thus been explored and investigated. For each parameter introduced, the minimum cardinality of certain standard classes of graphs, bounds, and characterization results was explored.