A novel approach to determining chromatic number induced by labelings

Given a simple graph $G=(V,E)$ of order $p$ and size $q$, a bijection $f : V\cup E \to \{1, 2, \ldots, p+q\}$ is a local total neighborhood antimagic labeling of $G$ if the induced vertex coloring has the property $f^+_{tn}(u) \ne f^+_{tn}(v)$ for every two adjacent vertices $u$ and $v$ where $f^+_{tn}(u) = \sum (f(ux) + f(x))$ over every neighbor $x$ of $u$. The local total neighborhood antimagic chromatic number of $G$, denoted $χ_{ltna}(G)$ is the minimum number of distinct induced colors over all local total neighborhood antimagic labeling of $G$. In this paper, we determine the local total neighborhood antimagic chromatic number of the join of graphs with distinct parity orders.