Strongly Antimagic Labelling on Disjoint Union of Star and Cycle Graphs

Heng, Gene Sern (2026) Strongly Antimagic Labelling on Disjoint Union of Star and Cycle Graphs. Final Year Project (Bachelor), Tunku Abdul Rahman University of Management and Technology.

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Abstract

An antimagic labelling for a graph G with m edges is a bijection f(G)→{1,2,...,m} such that φf(u) ≠ φf(v) holds for any pair of distinct vertices u,v ∈ V(G), where φf(x) = Σf(e), x ∈ e. A strongly antimagic labelling is an antimagic labelling with an additional condition: For any u,v ∈ V(G), if deg(u) > deg(v), then φf(u) > φf(v). A graph G is strongly antimagic if it admits a strongly antimagic labelling. Many graphs have been shown to be antimagic but have not been proved to be strongly antimagic. The disjoint union of a graph, also known as the graph sum, is the combination of two or more graphs to form a larger graph, while all graphs are disconnected. Let n ≥ 3 be an integer. We prove that n-star and 4-cycle disjoint union graphs, n-star and n-cycle disjoint union graphs and 2 copies of n-star and n-cycle disjoint union graphs are strongly antimagic.