On local antimagic chromatic number of lexicographic product graphs

Abstract

Let $G=(V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic if $G$ admits a local antimagic labeling. A bijection $f:E\to \{1,2,\dots ,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $f^{+}(u)\ne f^{+}(v)$, where $f^{+}(u)={\sum }_{e\in E(u)}f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $f^{+}(v)$. The local antimagic chromatic number, denoted ${\chi }_{la}(G)$, is the minimum number of induced colors taken over local antimagic labeling of $G$. Let $G$ and $H$ be two vertex disjoint graphs. The {\it lexicographic product} of $G$ and $H$, denoted $G[H]$, is the graph with vertex set $V(G)\times V(H)$, and $(u,u^{\prime })$ is adjacent to $(v,v^{\prime })$ in $G[H]$ if $(u,v)\in E(G)$ or if $u=v$ and $u^{\prime }{v}^{\prime }\in E(H)$. In this paper, we obtained sharp upper bound of ${\chi }_{la}(G[O_n])$ where $O_n$ is a null graph of order $n\ge 1$. Sufficient conditions for even regular bipartite and tripartite graphs $G$ to have ${\chi }_{la}(G)=3$ are also obtained. Consequently, we successfully determined the local antimagic chromatic number of infinitely many (connected and disconnected) regular graphs that partially support the existence of $r$-regular graph $G$ of order $p$ such that (i) ${\chi }_{la}(G)=\chi (G)=k$, and (ii) ${\chi }_{la}(G)=\chi (G)+1=k$ for each possible $r,p,k$.