Affirmative Solutions on Local Antimagic Chromatic Number

An edge labeling of a connected graphG = (V,E) is said to be local antimagic if it is a bijectionf:E ->{1, horizontal ellipsis ,|E|} such that for any pair of adjacent verticesxandy,f+(x)not equal f+(y), where the induced vertex labelf+(x)= n-ary sumation f(e), witheranging over all the edges incident tox. 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 proves O2) 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 proves On) 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.