Opuscula Math. 43, no. 3 (2023), 429-453
https://doi.org/10.7494/OpMath.2023.43.3.429

Opuscula Mathematica

# On local antimagic total labeling of complete graphs amalgamation

Gee-Choon Lau
Wai Chee Shiu

Abstract. 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 $$g : E \to \{1,2,\ldots,q\}$$ is called a local antimagic labeling of \$ if for any two adjacent vertices $$u$$ and $$v$$, we have $$g^+(u) \ne g^+(v)$$, where $$g^+(u) = \sum_{e\in E(u)} g(e)$$, and $$E(u)$$ is the set of edges incident to $$u$$. Similarly, a bijection $$f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}$$ is called a local antimagic total labeling of $$G$$ if for any two adjacent vertices $$u$$ and $$v$$, we have $$w_f(u)\ne w_f(v)$$, where $$w_f(u) = f(u) + \sum_{e\in E(u)} f(e)$$. 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 $$\chi_{la}(G)$$ (respectively $$\chi_{lat}(G)$$), is the minimum number of induced colors taken over local antimagic (total) labeling of $$G$$. In this paper, we determined $$\chi_{lat}(G)$$ where $$G$$ is the amalgamation ofcomplete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of $$K_1$$ and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.

Keywords: local antimagic (total) chromatic number, amalgamation, complete graphs.

Mathematics Subject Classification: 05C78, 05C15.

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• Gee-Choon Lau (corresponding author)
• https://orcid.org/0000-0002-9777-6571
• Universiti Teknologi MARA (Segamat Campus), College of Computing, Informatics & Media, 85000 Johor, Malaysia
• Communicated by Andrzej Żak.
• Revised: 2023-03-27.
• Accepted: 2023-03-28.
• Published online: 2023-05-17.