The upper bound of vertex local antimagic edge labeling on graph operations

Abstract

Let G(V(G), E(G)) be a connected simple graph and let u, v be two vertices of graph G. A bijective function f from the edge set of G to the natural number up to the number of edges in G is called a vertex local antimagic edge labeling if for any two adjacent vertices v and v′, w(v) = w(v′), where , and E(v) is the set of vertices which is incident to v. Thus any vertex local antimagic edge labeling induces a proper vertex coloring of G where the vertex v is assigned with the color w(v). The vertex local antimagic chromatic number χla (G) is the minimum number of colors taken over all coloring induced by vertex local antimagic edge labeling of G. In this paper, we study the vertex local antimagic edge labeling of graphs and determine the upper bound of vertex local antimagic chromatic number on graph operation, namely cartesian product of graphs, corona product of graphs, and power graph.