Local Total Antimagic Chromatic Number for the Disjoint Union of Star Graphs (European Journal of Pure and Applied Mathematics, 2024)
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 $f:V(G)\cup E(G)→ \{ 1,2,...,n+m\}$, with for any two adjacent vertices $u$ and $v$ with weights $w(u) eq w(v)$, for any two adjacent edges $e_1=uv$ and $e_2=vw$ with their weights $w(e) eq w(e')$ and any vertex $x$ incident to an edge $e=xy$ with their weights $w(x) eq w(xy)$. 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 $χ_{lt}(G)$. In this paper, we determine the local total antimagic chromatic number for the disjoint union of star graphs.
Keywords: Local antimagic graphs, chromatic number, total coloring