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)) →{1, 2,...,k} as a vertex irregular k-labeling and w : V (G) → N, for every uv ∈ E(G), w(u)≠w(v) where w(u) = ∑ v∈N(u)l(i) and (ii) opt(l) = min{max{li : li is a vertex irregular labeling}}. The chromatic number of the local irregularity vertex coloring of G denoted by χlis(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 Pm⊙ G when G is a family of tree graphs, centipede Cn, double star graph (S2,n), Weed graph (S3,n), and E graph (E3,n).