On local antimagic total labeling of complete graphs amalgamation

• Autorzy: Lau Gee-Choon , Shiu Wai-Chee

Identyfikatory

DOI

10.7494/OpMath.2023.43.3.429

• Języki publikacji: EN

Abstrakty

EN

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) = ∑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. 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.

Słowa kluczowe

EN

local antimagic (total) chromatic number; amalgamation; complete graph

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2023
• Tom: Vol. 43, no. 3
• Strony: 429--453
• Opis fizyczny: Bibliogr. 14 poz., tab.

Twórcy

autor

Lau Gee-Choon

• Universiti Teknologi MARA (Segamat Campus), College of Computing, Informatics & Media, 85000 Johor, Malaysia

• https://orcid.org/0000-0002-9777-6571

autor

Shiu Wai-Chee

• The Chinese University of Hong Kong, Department Mathematics, Shatin, Hong Kong

• https://orcid.org/0000-0002-2819-8480

Bibliografia

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• [2] J. Bensmail, M. Senhaji, K.S. Lyngsie, On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture, Discrete Math. Theoret. Comput. Sci. 19 (2017), no. 1, #22.
• [3] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, New York, MacMillan, 1976.
• [4] T.R. Hagedorn, Magic rectangles revisited, Discrete Math. 207 (1999), 65–72.
• [5] J. Haslegrave, Proof of a local antimagic conjecture, Discrete Math. Theor. Comput. Sci. 20 (2018), no. 1, #18.
• [6] G.C. Lau, J. Li, H.K. Ng, W.C. Shiu, Approaches which output infinitely many graphs with small local antimagic chromatic number, Disc. Math. Algorithms Appl. 15, no. 2, 2250079 (2023).
• [7] G.C. Lau, H.K. Ng, W.C. Shiu, Affirmative solutions on local antimagic chromatic number, Graphs Combin. 36 (2020), 1337–1354.
• [8] G.C. Lau, K. Schaeffer, W.C. Shiu, Every graph is local antimagic total and its applications (2022), preprint.
• [9] G.C. Lau, W.C. Shiu, H.K. Ng, On local antimagic chromatic number of graphs with cut-vertices, Iran. J. Math. Sci. Inform. (2022), accepted.
• [10] G.C. Lau, W.C. Shiu, H.K. Ng, On local antimagic chromatic number of cycle-related join graphs, Discuss. Math. Graph Theory 41 (2021), 133–152.
• [11] G.C. Lau, W.C. Shiu, C.X. Soo, On local antimagic chromatic number of spider graphs, J. Discrete Math. Sci. Cryptogr. (2022), published online.
• [12] K. Premalatha, S. Arumugam, Y-C. Lee, T.-M. Wang, Local antimagic chromatic number of trees – I, J. Discrete Math. Sci. Cryptogr. 25 (2022), no. 6, 1591–1602.
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Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)