Super Total Local Antimagic Vertex Coloring of Graphs

Let $G=(V,E)$ be a finite simple undirected graph without isolated vertices. A bijective map $f:V\cup E\to \{1,2,\dots ,\mathrm{\mid }V\mathrm{\mid }+\mathrm{\mid }E\mathrm{\mid }\}$ is called total local antimagic labeling if for each edge $uv\in E,w(u)\mathrm{\ne }w(v)$, where $w(v)$ is a weight of a vertex $v$ defined by $w(v)={\sum }_{x\in NT(v)}f(x)$, where $NT(u)=N(u)\cup \{uv:uv\in E\}$ is the total open neighborhood of a vertex $u$. Further, $f$ is called super vertex total local antimagic labeling or super edge total local antimagic labeling if $f(V)=\{1,2,\dots ,\mathrm{\mid }V\mathrm{\mid }\}$ or $f(E)=\{1,2,\dots ,\mathrm{\mid }E\mathrm{\mid }\}$, respectively. The labeling $f$ induces a proper vertex coloring of $G$. The super vertex (edge) total local antimagic chromatic number of a graph $G$ is the minimum number of colors used overall colorings of $G$ induced by super vertex (edge) total local antimagic labeling of $G$. In this paper, we have calculated the super vertex (edge) total local antimagic chromatic number of some families of graphs.