On Local Antimagic b-Coloring of Graphs: New Notion

Abstract

Let $G=(V,E)$ be a simple, connected and un-directed graph. Given that a map $f: E(G) \longrightarrow \{1,2,3, \dots, |E(G)|\}$. We define a vertex weight of $v\in V$ as $w(v)=\Sigma_{e\in E(v)}f(e)$ where $E(v)$ is the set of edges incident to $v$. The bijection $f$ is said to be a local antimagic labeling if for any two adjacent vertices, their vertex weights must be distinct. Furthermore a $b-$coloring of a graph is a proper $k-$coloring of the vertices of $G$ such that in each color class there exists a vertex having neighbors in all other $k-1$ color classes. If we assign color on each vertex by the vertex weight $w(v)$ such that it induces a graph coloring satisfying $b-$coloring property, then this concept falls into a local antimagic $b-$coloring of graph. A local antimagic $b-$chromatic number, denoted by $\varphi_{la}(G),$ is the maximum number of colors chosen for any colorings generated by local antimagic $b-$coloring of $G$. In this study, we initiate to study the $b-$chromatic number of $G$ and the exact values of $\varphi_{la}(G)$ of certain classes of graph families.