Local Total Antimagic Chromatic Number for the Disjoint Union of Star Graphs

Abstract

Let $G$ be a graph with $n$ vertices and $m$ edges without  isolated vertices.
A local total antimagic labeling of a graph $G$ is defined as there is a bijection $f:V(G)\cup E(G)\to \{1,2,...,n+m\}$, with for any two adjacent vertices $u$ and $v$ with weights $w(u)\ne w(v)$, for any two adjacent edges $e_1=uv$  and $e_2=vw$ with their weights  $w(e)\ne w(e^{\prime })$ and any vertex $x$ incident to an edge $e=xy$ with their weights $w(x)\ne w(xy)$.  The vertex weight $w(u)$ is defined by $w(u)=\sum _{e\in E(u)}f(e)$, where $E(u)$ is the set of edges incident to $u$.  The edge weight $w(e=pq)$ is defined by  $w(e=pq)=f(p)+f(q)$.  The local total antimagic chromatic number is the minimum number of colors taken over all induced by local total antimagic colorings (labelings) of $G$, which is denoted by  ${\chi }_{lt}(G)$. In this paper, we determine the local total antimagic chromatic number for the disjoint union of star graphs.