$b$-Local Irregular Chromatic Number of Graphs

In this paper, we study a new notation of coloring of graph, namely a $b$-local irregularity coloring. Suppose $l : V(G) \rightarrow \{1,2, \ldots ,k\}$ is called vertex irregular $k$-labeling and $w : V(G) \rightarrow N$, where $w(u) = \sum_{v \in N(u)} l(v)$. Every color class has a representative adjacent to at least one vertex in each of the color classes.$l$ is $b$-local irregularity coloring. The $b$-local irregular chromatic number denoted by $\chi_{b-lis}(G)$ is the largest of $k$ such that $G$ admits a $b$-local irregularity coloring. In this paper, we study the $b$-local irregular chromatic number of graphs namely path, cycle, star, friendship, complete, complete bipartite, and Wheel graph.