Local Antimagic Chromatic Number of Middle Graph and Direct Product of Graphs

An edge labeling of a connected graph $G=(V,E)$ is called a local antimagic labeling if it is a bijection $f:E\to \{1,2,\cdots ,\mid E\mid \}$ such that for any pair of adjacent vertices $u$ and $v$, $f^{+}(u)\mathrm{\ne }{f}^{+}(v)$ where $f^{+}(x)=\sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted ${\chi }_{la}(G)$, is the minimum number of distinct induced vertex labels across all possible local antimagic labelings of $G$. This paper investigates two families of graphs $G(m^{[n]})$ (a graph class constructed by augmenting a bipartite structure $B(m^{[n]})$ with cyclic edges) and $K_{1,s}\times C_n$ (the direct product of a star $K_{1,s}$ and a cycle $C_n$). Under the conditions odd integer $m\ge 1$ and even integer $n\ge 4$, we establish the exact values of ${\chi }_{la}(G(m^{[n]}))$ and ${\chi }_{la}(K_{1,s}\times C_n)$ for various $s$. Furthermore, we characterize several graph families achieving ${\chi }_{la}(G)=2$ or $3$.