Affirmative Solutions on Local Antimagic Chromatic Number - Lau, Gee-Choon; Ng, Ho-Kuen; Shiu, Wai-Chee (Graphs and combinatorics, 2020)
An edge labeling of a connected graph G = ( V , E ) is said to be local antimagic if it is a bijection f : E → { 1 , … , | E | } such that for any pair of adjacent vertices x and y , f + ( x ) ≠ f + ( y ) , where the induced vertex label f + ( x ) = ∑ f ( e ) , with e ranging over all the edges incident to x . The local antimagic chromatic number of G , denoted by χ 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 χ la ( G ∨ O 2 ) that was obtained in [Local antimagic vertex coloring of a graph, Graphs Combin. 33:275–285 (2017)]. A sharp lower bound of χ la ( G ∨ 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.