The THE LOCAL ANTIMAGIC TOTAL CHROMATIC NUMBERS ON BARBELL WHEEL GRAPHS

Abstract

\textbf{Abstract}. % Dalam bahasa Inggris \textit{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 $\{1,2,\dots,|V(G)|+|E(G)|\}$ such as for all two adjacent vertices $u$ and $v$ we have $w_t(u)\neq w_t(v)$, where $w_t(u)=f(u)+\sum_{e\in E(u)}{f(e)}$ 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 $\chi_{lat}(G)$. The barbell wheel graph $BW_{n,k}$, with $n\geq3$ and $k\geq2$, is defined as a graph with two subgraphs of wheels $W_n$ that are connected by the path subgraph $P_k$ at each center vertex. In this paper, we prove that the barbell wheel graph $BW_{n,k}$ has local antimagic total labeling. We also determine its local antimagic total chromatic number.}\\