ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH

Abstract

Let \(G=(V,E)\) be a graph with a vertex set \(V\) and an edge set \(E\). The graph \(G\) is said to be with a local irregular vertex coloring if there is a function \(f\) called a local irregularity vertex coloring with the properties: (i) \(l:(V(G)) \to \{ 1,2,...,k \} \) as a vertex irregular \(k\)-labeling and \(w:V(G)\to N,\) for every \(uv \in E(G),\) \({w(u)\neq w(v)}\) where \(w(u)=\sum_{v\in N(u)}l(i)\) and  (ii) \(\mathrm{opt}(l)=\min\{ \max \{ l_{i}:  l_{i} \ \text{is a vertex irregular labeling}\}\}\). The chromatic number of the local irregularity vertex coloring of \(G\) denoted by \(\chi_{lis}(G)\), is the minimum cardinality of the largest label over all such local irregularity vertex colorings. In this paper, we study a local irregular vertex coloring of \(P_m\bigodot G\) when \(G\) is a family of tree graphs, centipede \(C_n\), double star graph \((S_{2,n})\), Weed graph \((S_{3,n})\), and \(E\) graph \((E_{3,n})\).