On local antimagic chromatic number of lexicographic product graphs

Consider a simple connected graph $$G = (V,E)$$ of order p and size q. For bijection $$f : E \to \{1,2,\ldots,q\}$$ , let $$f^+(u) \sum_{e\in E(u)} f(e)$$ where $$E(u)$$ is the set edges incident to u. We say f local antimagic labeling G if for any two adjacent vertices u v, we have \ne f^+(v)$$ . The minimum number distinct values $$f^+$$ taken over all denoted by $$\chi_{la}(G)$$ Let $$G[H]$$ be lexicographic product graphs H. In this paper, obtain sharp upper bound $$\chi_{la}(G[O_n])$$ $$O_n$$ null $$n\ge 3$$ Sufficient conditions even regular bipartite tripartite $$\chi_{la}(G)=3$$ are also obtained. Consequently, successfully determined chromatic infinitely many (connected disconnected) that partially support existence an r-regular such (i) $$\chi_{la}(G)=\chi(G)=k$$ (ii) $$\chi_{la}(G)=\chi(G)+1=k$$ each possible $$r,p,k$$