Local total neighborhood antimagic labelings–a new variant of vertex colorings

Let $G=(V,E)$ be a simple graph of order $p$ and size $q$. A local total neighborhood antimagic labeling of $G$ is a bijection $f:V\cup E\to [1,p+q]$ such that, for every vertex $u\in V$, the induced vertex color $f_{tn}^{+}(u)=\sum _{ux\in E}[f(ux)+f(x)]$ assigns distinct integers (called colors) to adjacent vertices of $G$. We define $f_{tn}^{+}(u)=0$ when $u$ is an isolated vertex. A graph $G$ is said to be local total neighborhood antimagic if it admits a local total neighborhood antimagic labeling. The local total neighborhood antimagic chromatic number of $G$, denoted by ${\chi }_{ltna}(G)$, is the minimum number of distinct induced colors taken over all such labelings of $G$. In this paper, we investigated properties of ${\chi }_{ltna}(G)$ for graphs without isolated vertices and provide results that characterize this chromatic parameter. We focused on several graph classes, including paths, cycles, complete graphs, complete bipartite graphs, wheel graphs, and various types of join graphs.