A note on local antimagic chromatic number of lexicographic product graphs

Abstract

Let $G=(V,E)$ be a connected simple graph. A bijection $f:E\to \{1,2,\dots ,|E|\}$ is called a local antimagic labeling of $G$ if $f^{+}(u)\ne f^{+}(v)$ holds for any two adjacent vertices $u$ and $v$, where $f^{+}(u)={\sum }_{e\in E(u)}f(e)$ and $E(u$) is the set of edges incident to $u$. A graph $G$ is called local antimagic if $G$ admits at least a local antimagic labeling. The local antimagic chromatic number, denoted ${\chi }_{la}(G)$, is the minimum number of induced colors taken over local antimagic labelings of $G$. Let $G$ and $H$ be two disjoint graphs. The graph $G[H]$ is obtained by the lexicographic product of $G$ and $H$. In this paper, we obtain sufficient conditions for ${\chi }_{la}(G[H])\le {\chi }_{la}(G){\chi }_{la}(H)$. Consequently, we give examples of $G$ and $H$ such that ${\chi }_{la}(G[H])=\chi (G)\chi (H)$, where $\chi (G)$ is the chromatic number of $G$. We conjecture that (i) there are infinitely many graphs $G$ and $H$ such that ${\chi }_{la}(G[H])={\chi }_{la}(G){\chi }_{la}(H)=\chi (G)\chi (H)$, and (ii) for $k\ge 1$, ${\chi }_{la}(G[H])=\chi (G)\chi (H)$ if and only if $\chi (G)\chi (H)=2\chi (H)+\lceil \frac{\chi (H)}{k}\rceil$, where $2k+1$ is the length of a shortest odd cycle in $G$.