on local antimagic chromatic number of graphs

A local antimagic labeling of a connected graph g with at least three vertices, is a bijection f : e(g) → {1, 2, . . . , |e(g)|} such that for any two adjacent vertices u and v of g, the condition ωf (u) ̸= ωf (v) holds; where ωf (u) = ∑ x∈n(u) f(xu). assigning ωf (u) to u for each vertex u in v (g), induces naturally a proper vertex coloring of g; and |f| denotes the number of colors appearing in this proper vertex coloring. the local antimagic chromatic number of g, denoted by χla(g), is defined as the minimum of |f|, where f ranges over all local antimagic labelings of g. in this paper, we explicitly construct an infinite class of connected graphs g such that χla(g) can be arbitrarily large while χla(g∨ k¯ 2) = 3, where g ∨ k¯ 2 is the join graph of g and the complement graph of k2. the aforementioned fact leads us to an infinite class of counterexamples to a result of [local antimagic vertex coloring of a graph, graphs and combinatorics 33 (2017), 275–285].