Antimagic and Prime Labelling on Moser Spindle and Its Related Graphs

Tan, Jin Min (2025) Antimagic and Prime Labelling on Moser Spindle and Its Related Graphs. Final Year Project (Bachelor), Tunku Abdul Rahman University of Management and Technology.

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Abstract

Antimagic labelling for a graph G is defined as assigning each edge a unique positive integer and labeling each vertex with the sum of its adjacent edge labels. The condition is that all vertex labels must be distinct. Prime labelling involves assigning the integers 1 through the number of vertices of a graph G such that every pair of adjacent vertices is coprime. A related labelling is coprime labelling, where integers 1 through pr(G) are assigned, with pr(G) being the minimum value that allows such a labelling. This study explores antimagic labelling, prime labelling, and coprime labelling of the Moser Spindle graph and several of its transformations, including its line graph, complement graph, dual graph, 2-distance graph, and subdivision graph. Through a series of propositions, we attempt to prove that all these graphs admit an antimagic labelling, supporting the longstanding Hartsfield and Ringel conjecture. We also analyze the potential for prime labellings across these graph forms. For graphs that do not accept prime labelling, we determine the minimum value pr(G) that allows graph G to accept a coprime labelling. These findings contribute new insights into the labelling characteristics of well-known graph structures and their transformations. Keywords: antimagic labelling, prime labelling, coprime labelling, Moser Spindle, line graph, complement graph, dual graph, 2-distance graph, subdivision graph