On the local antimagic labeling of graphs amalgamation

Abstract

Let G be a simple connected graph, an ordered pair of sets G(V, E), with V is a set of vertices and E is a set of edges. Graph coloring has been one of the most popular branches in topics of graph theory. In 2017 Arumugam et al. developed a new notion of coloring, namely local antimagic coloring of a graph. This concept is a combination of graph labeling and graph coloring. The local antimagic of graphs is one of the colorings of graph theory that is interesting to study. By the definition, the local antimagic coloring is a bijection f : E(G) ƒ> {1, 2, 3,…, |E(G)|} if for any two adjacent vertices u and v, w(u) = w(v), where w(v) = ΣeeE(v) 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 xla(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we will study the local antimagic coloring of amalgamation of graphs.