On the local antimagic chromatic number of the lexicographic product of graphs

Let G = (V, E) be a connected simple graph. A bijection f : E -> {1, 2, . . . , |E|} is said to be a local antimagic labeling of G if f(+)(u) not equal f(+)(v) holds for any two adjacent vertices u and v of G, where E(u) is the set of edges incident to u and f(+)(u) = Sigma(eE(u)) f(e). A graph G is called local antimagic if G admits at least one 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]) <= 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 >= 1, chi(la)(G[H]) = chi(G)chi(H) if and only if chi(G)chi(H) = 2 chi(H) + inverted right perpendicular chi(H)/k inverted left perpendicular, where 2k + 1 is the length of a shortest odd cycle in G.