A novel approach to determining chromatic number induced by labelings

Abstract

Given a simple graph $G=(V,E)$ of order $p$ and size $q$, a bijection $f:V\cup E\to \{1,2,\dots ,p+q\}$ 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 ${\chi }_{ltna}(G)$ 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.

arXiv commentsubmitted to Acta Mathematica Hungarica