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 $f: V \cup E \rightarrow \{1,2, \dots, |V|+ |E| \}$ 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.