For every finite connected graph G=(V,E) with G notcong K_2, there exists a bijection f:E→1,2,…,|E| such that the vertex-sums s_f(v)=∑_e∈ E:,v∈ ef(e) are pairwise distinct; that is, s_f(u)≠ s_f(v) for all distinct vertices u,v∈ V.

Hartsfield–Ringel antimagic conjecture

Hartsfield and Ringel conjectured that every connected graph other than $K_2$ has an antimagic labeling, meaning all vertex incident-edge sums are distinct. The conjecture remains unresolved for arbitrary connected graphs.

Known results

Paths, stars, cycles, complete graphs, wheels, and $K_{2,m}$ for $m \ge 3$ are antimagic.
Graphs with minimum degree at least $C\log n$ and complete multipartite graphs other than $K_2$ are antimagic (2003).
Graphs with sufficiently large average degree are antimagic when isolated vertices and isolated edges are absent (2014).
Regular graphs and several further tree classes are known cases; the local antimagic conjecture is weaker and does not settle this problem.

September 28, 2026 claimed join-graph advance

A preprint by Grégoire Beaudoire, Cédric Bentz, and Christophe Picouleau claims that every join graph with at least three vertices admits an antimagic labeling. This establishes a broad new family if correct, but it does not prove the full conjecture and has not been independently verified in the retrieved sources.

Current status (as of September 2026): The conjecture is proved for many graph classes and is claimed for all join graphs with at least three vertices, but it remains open for arbitrary connected graphs other than $K_2$ .

Hartsfield–Ringel antimagic conjecture

Primary source

Additional references

Progress summary

Solutions 0