An edge labeling of a connected graph \(G = (V, E)\) is said to be local antimagic if it is a bijection \(f:E \rightarrow \{1,\ldots ,|E|\}\) such that for any pair of adjacent vertices x and y, \(f^+(x)\not = f^+(y)\), where the induced vertex label \(f^+(x)= \sum f(e)\), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by \(\chi _{la}(G)\), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we give counterexamples to the lower bound of \(\chi _{la}(G \vee O_2)\) that was obtained in [Local antimagic vertex coloring of a graph, Graphs Combin. 33:275–285 (2017)]. A sharp lower bound of \(\chi _{la}(G\vee O_n)\) and sufficient conditions for the given lower bound to be attained are obtained. Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative. We also completely determined the local antimagic chromatic number of complete bipartite graphs.

如果连接图\（G =（V，E）\）的边缘标记是双射\\（f：E \ rightarrow \ {1，\ ldots，| E | \} \），则被认为是局部反魔术的。这样对于任意一对相邻顶点x和y，\（f ^ +（x）\ not = f ^ +（y）\），其中诱导的顶点标签\（f ^ +（x）= \ sum f（ e）\），其中e覆盖入射到x的所有边。G的局部反色数，用\（\ chi _ {la}（G）\）表示，是G的所有局部反色标记上不同的诱导顶点标记的最小数量。在本文中，我们给出了下限的反例\（\ chi _ {la}（G \ vee O_2）\）是在[图形的局部反魔术顶点着色，图形组合]中获得的。33：275–285（2017）]。获得\（\ chi _ {la}（G \ vee O_n）\）的尖锐下界，并获得了达到给定下界的充分条件。此外，我们确定了定理2.15，并肯定地解决了问题3.3。我们还完全确定了完整二部图的局部反色数。