On the local irregularity vertex coloring of volcano, broom, parachute, double broom and complete multipartite graphs

Abstract

Let G = (V,E) be a simple, finite, undirected, and connected graph with vertex set V (G) and edge set E(G). A bijection l : V (G) → {1, 2,...,k} is label function l if opt(l) = min{max(li) : li vertex irregular labeling} and for any two adjacent vertices u and v, w(u) ̸= w(v) where w(u) = P v∈N(u) l(v) and N(u) is set of vertices adjacent to v. w is called local irregularity vertex coloring. The minimum cardinality of local irregularity vertex coloring of G is called chromatic number local irregular denoted by χlis(G). In this paper, we verify the exact values of volcano, broom, parachute, double broom and complete multipartite graphs.