The THE LOCAL ANTIMAGIC TOTAL CHROMATIC NUMBERS ON BARBELL WHEEL GRAPHS

Abstract. % Dalam bahasa Inggris Let $G=(V,e)$ be a graph with a finite non-empty vertex set $V(G)$ and a edge set $E(G)$. A local antimagic total labeling on graph G defined as a bijective mapping f from a union of the vertex set and the edges set of G to a set of integers \1,2,,|V(G)|+|E(G)|\ such as for all two adjacent vertices u and v we have wₜ(u)≠ wₜ(v), where wₜ(u)=f(u)+∑e∈ E(u)f(e) is a weight of vertex u, and $E(u)$ is a set of adjacent edges on the vertex u. Each distinct vertex weight in local antimagic total labeling can be considered as distinct colors, so that local antimagic total labeling on graph G induces vertex coloring on graph G, with minimum numbers of colors or its chromatic number denoted as χₗₐₜ(G). The barbell wheel graph BWn,k, with n≥3 and k≥2, is defined as a graph with two subgraphs of wheels Wₙ that are connected by the path subgraph Pₖ at each center vertex. In this paper, we prove that the barbell wheel graph BWn,k has local antimagic total labeling. We also determine its local antimagic total chromatic number.\\