Local Antimagic Vertex Coloring of a Graph

Abstract

Let $$G=(V,E)$$G=(V,E) be a connected graph with $$\left| V \right| =n$$V=n and $$\left| E \right| = m.$$E=m. A bijection $$f:E \rightarrow \{1,2, \dots , m\}$$f:E?{1,2,?,m} is called a local antimagic labeling if for any two adjacent vertices u and v, $$w(u)\ne w(v),$$w(u)?w(v), where $$w(u)=\sum \nolimits _{e\in E(u)}{f(e)},$$w(u)=?e?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 $$\chi _{la}(G)$$?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.