Local Antimagic Vertex Coloring of a Graph

Abstract

Let $G=(V,E)$ be a connected graph with $\left| V \right| =n$ and $\left| E \right| = m.$ A bijection $f:E \rightarrow \{1,2, \dots , m\}$ is called a local antimagic labeling if for any two adjacent vertices u and v,  $w(u)\ne w(v),$ where $w(u)=\sum \nolimits _{e\in E(u)}{f(e)},$ and E(u) is the set of edges incident to u. Thus any local antimagic labeling induces a proper vertex coloring of G where the vertex v is assigned the color w(v). The local antimagic chromatic number $\chi _{la}(G)$ is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper we present several basic results on this new parameter.