Every graph is local antimagic total and its applications

Let $ G = (V, E) $ be a simple graph of order $ p $ and size $ q $. A graph $ G $ is called local antimagic (total) if $ G $ admits a local antimagic (total) labeling. A bijection $ g : E → {1, 2, . . . , q} $ is called a local antimagic labeling of $ G $ if for any two adjacent vertices $ u $ and $ v $, we have $g^+(u) ≠ g^+(v) $, where $ g^+(u) = \Sigma_{e∈E(u)} g(e) $, and $ E(u) $ is the set of edges incident to $ u $. Similarly, a bijection $f : V (G)∪E(G) → {1, 2, . . . , p+q} $ is called a local antimagic total labeling of $ G $ if for any two adjacent vertices $ u $ and $ v $, we have $ w_f (u) ≠ w_f (v) $, where $ w_f (u) = f(u) + \Sigma_{e∈E(u)} f(e) $. Thus, any local antimagic (total) labeling induces a proper vertex coloring of $ G $ if vertex $ v $ is assigned the color $ g^+ (v) $ (respectively, $ w_f (u) $). The local antimagic (total) chromatic number, denoted $ χ_{la} (G) $ (respectively $ χ_{lat} (G)$ ), is the minimum number of induced colors taken over local antimagic (total) labeling of $ G $. We provide a short proof that every graph $ G $ is local antimagic total. The proof provides sharp upper bound to $ χ_{lat} (G) $. We then determined the exact $ χ_{lat} (G) $, where $ G $ is a complete bipartite graph, a path, or the Cartesian product of two cycles. Consequently, the $ χ_{la} (G ∨ K_1) $ is also obtained. Moreover, we determined the $ χ_{la} (G ∨ K_1) $ and hence the $χ_{lat} (G) $ for a class of 2-regular graphs $ G $ (possibly with a path). The work of this paper also provides many open problems on $ χ_{lat} (G) $. We also conjecture that each graph $ G $ of order at least 3 has $ χ_{lat} (G) ≤ χ_{la} (G) $.