EVERY GRAPH IS LOCAL ANTIMAGIC TOTAL AND ITS APPLICATIONS

• Published in: Opuscula Mathematica
• Main Author: Lau G.-C.; Schaffer K.; Shiu W.C.
• Format: Article
• Language: English
• Published: AGH University of Science and Technology 2023
• Online Access: https://www.scopus.com/inward/record.uri?eid=2-s2.0-85170201438&doi=10.7494%2fOpMath.2023.43.6.841&partnerID=40&md5=24ba28f4666342199f729da3b72c2d07

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) = ∑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 wf(u) ≠ wf(v), where wf(u) = f(u) + ∑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, wf(u)). The local antimagic (total) chromatic number, denoted χla(G) (respectively χlat(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 χlat(G). We then determined the exact χlat(G), where G is a complete bipartite graph, a path, or the Cartesian product of two cycles. Consequently, the χla(G ∨ K1) is also obtained. Moreover, we determined the χla(G ∨ K1) and hence the χlat(G) for a class of 2-regular graphs G (possibly with a path). The work of this paper also provides many open problems on χlat(G). We also conjecture that each graph G of order at least 3 has χlat(G) ≤ χla(G). © 2023 Authors. Creative Commons CC-BY 4.0.