An Inclusive Local Irregularity Coloring of Graphs

Abstract

All graph in this paper are connected and simple. Let G = (V, E) be a simple graph, where V (G) is vertex set and E(G) is edge set. The local irregularity vertex coloring of G is l : V (G) → {1, 2, · · · , k} and w : V (G) → N where w(u) = Σv∈N(u) l(v) such that opt(l) = min{max{li} and for every uv ∈ E(G), w(u) 6= w(v), w is a local irregularity vertex coloring. The minimum of color set is called the local irregular chromatic number, denoted by χlis(G). In this paper, we determine the local irregular chromatic number of graphs.