Local Distance Antimagic Labeling of Neighborhood Balanced Colored Graphs

Abstract

Let $G=(V,E)$ be a graph without isolated vertices. A bijection $f:V\to \{1,2,\dots ,|V|\}$ is called a local distance antimagic labeling if $w(u)\ne w(v)$ for every edge $uv\in E$, where the weight $w(u)$ of a vertex $u$ is defined as the sum of labels assigned to all vertices adjacent to $u$. The local distance antimagic chromatic number of $G$, denoted by ${\chi }_{ld}(G)$, is defined as the minimum number of colors taken over all colorings of $G$ induced by local distance antimagic labelings. In this paper, we investigate local distance antimagic labelings of the direct product and the union of certain classes of graphs, namely, neighborhood balanced $k$-colored graphs and nearly neighborhood balanced $k$-colored graphs.