Super Total Local Antimagic Vertex Coloring of Graphs

Abstract

Let $G=(V,E)$ be a finite simple undirected graph without isolated vertices. A bijective map $f:V\cup E\to \{1,2,\dots ,|V|+|E|\}$ is called total local antimagic labeling if for each edge $uv\in E,w(u)\ne w(v)$, 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 $f(V)=\{1,2,\dots ,|V|\}$ or $f(E)=\{1,2,\dots ,|E|\}$, 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.

Rectified few bounds and calculated some exact values for the defines numbers