Antimagic Labeling in Graph Theory

Antimagic labeling is a distinguished branch of graph theory concerned with assigning distinct positive integers to the edges of a finite simple graph so that the sums of labels incident on each vertex—known as vertex sums—are pairwise distinct. Originating from a conjecture that every connected graph except the two-vertex path admits such a labelling, the field has evolved into a rich interplay of combinatorial design, algebraic methods and graph operations. Core advances have addressed broad classes of graphs—regular graphs, complete bipartite graphs and Cartesian products—often employing bijective constructions or flow techniques over abelian groups. Practical implications range from network security protocols, where unique vertex signatures aid in anomaly detection, to the design of communication topologies with guaranteed collision-free labelling. Computational methods now complement theoretical proofs, yielding explicit algorithms for certain graph families. Recent years have seen the integration of product operations—such as corona and rooted products—into the antimagic paradigm, refining our understanding of how global degree conditions influence labelling existence. This overview highlights both foundational results and emerging directions that collectively underscore the global significance of antimagic labelling in discrete mathematics and its applications.

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A recent study has explored the antimagicness of generalised edge corona graphs. By combining a base graph with attached subgraphs at each edge, the authors established algorithmic conditions under which the resulting construction admits an antimagic labelling. Their approach relies on a careful ordering of edges incident to vertices of maximum degree and ensures that vertex sums remain distinct across the composite structure.

Another contribution addressed product operations on regular graphs. It was shown that the rooted product and the standard corona product of two regular graphs admit antimagic labelings whenever the degree of the base graph meets or exceeds that of the attached graph. This result extends classical findings for complete and complete bipartite graphs to a broader family defined by balanced degree constraints.

A further work proposed an extension of the original conjecture to distance-based labellings, termed D-antimagic labellings. The authors conjectured necessary and sufficient conditions for a graph to admit such a labelling in terms of distinct neighbourhood patterns, verified the conjecture computationally for small orders and proved closure properties under disjoint union. This generalisation offers a unified framework that captures both classical antimagic labelling and its distance-oriented variants.

Antimagic Labeling in Graph Theory publication trend

The graph below shows the total number of articles in antimagic labeling in graph theory across all publications each year (not limited to Nature Index journals).

Technical terms

Antimagic labelling: An injective assignment of the integers 1 to |E| to the edges of a graph such that all vertex sums are pairwise distinct.

Vertex sum: The total of labels on all edges incident to a given vertex.

Regular graph: A graph in which every vertex has the same degree (number of incident edges).

Corona product: A graph operation formed by joining each vertex of a base graph to every vertex in a corresponding copy of a second graph.

References

1. On the antimagicness of generalized edge corona graphs. Heliyon (2024).
2. Antimagic Labeling for Product of Regular Graphs. Symmetry (2022).