ON LOCAL ANTIMAGIC TOTAL LABELING OF COMPLETE GRAPHS AMALGAMATION

• Published in: Opuscula Mathematica
• Main Author: 2-s2.0-85164321688
• Format: Article
• Language: English
• Published: AGH University of Science and Technology 2023
• Online Access: https://www.scopus.com/inward/record.uri?eid=2-s2.0-85164321688&doi=10.7494%2fOpMath.2023.43.3.429&partnerID=40&md5=8f225012c6d4875243b78e2d113392c1

Let G = (V, E) be a connected 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) = Pe∈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) + Pe∈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. In this paper, we determined χlat(G) where G is the amalgamation of complete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of K1 and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given. © 2023 Authors.