Let $G=(V,E)$ be a graph with vertex set V and edge set E. A local antimagic total vertex coloring f of a graph G with vertex-set V and edge-set E is an injective map from $V\cup E$ to $\{1,2,\dots ,|V|+|E\left|\right\}$ such that if for each $uv\in E\left(G\right)$ then $w\left(u\right)\ne w\left(v\right)$, where $w\left(u\right)={\sum }_{uv\in E\left(G\right)}f\left(uv\right)+f\left(u\right)$. If the range set f satisfies $f\left(V\right)=\{1,2,\dots ,|V\left|\right\}$, then the labeling is said to be local super antimagic total labeling. This labeling generates a proper vertex coloring of the graph G with the color $w\left(v\right)$ assigning the vertex v. The local super antimagic total chromatic number of graph G, ${\chi }_{lsat}\left(G\right)$ is defined as the least number of colors that are used for all colorings generated by the local super antimagic total labeling of G. In this paper we investigate the existence of the local super antimagic total chromatic number for some particular classes of graphs such as a tree, path, cycle, helm, wheel, gear, sun, and regular graphs as well as an amalgamation of stars and an amalgamation of wheels.

Let $G$ be a finite, connected, undirected, and simple graph and $W$ be a set of vertices in $G$. A representation multiset of a vertex $u$ in $V(G)$ with respect to $W$ is defined as the multiset of distances between $u$ and the vertices in $W$. If every two adjacent vertices in $V(G)$ have a distinct multiset representation, the set $W$ is called a local multiset resolving set of $G$. If $G$ has a local multiset resolving set, then this set with the smallest cardinality is called the local multiset basis, and its cardinality is the local multiset dimension of $G$. Otherwise, $G$ is said to have an infinite local multiset dimension. On the other hand, if every two adjacent vertices in $V(G) \backslash W$ have a distinct representation multiset, the set $W$ is called a local outer multiset resolving set of $G$. Such a set with the smallest cardinality is called the local outer multiset basis, and its cardinality is the local outer multiset dimension of $G$. Unlike the local multiset dimension, every graph has a finite local outer multiset dimension. This paper presents some basic properties of local (outer) multiset dimensions, including lower bounds for the dimensions and a necessary condition for a finite local multiset dimension. We also determine local (outer) multiset dimensions for some graphs of small diameter.