A labelling of the edges of a graph \(G\) with \(1, 2, \dots, |E(G)|\) is antimagic if the vertex-sums (the sum of incident edge-labels at each vertex) are pairwise distinct. Conjecture: every connected graph except \(K_2\) admits an antimagic labelling.
Conjectured by Nick Hartsfield and Gerhard Ringel in 1990, as the darker twin of the graceful-labelling story: where graceful labellings ask all edge-weights to be distinct within \(\{1, \dots, q\}\), antimagic labellings ask all vertex-sums to be pairwise distinct. The conjecture is verified for many families — paths, wheels, complete graphs, dense graphs (Alon and co-authors showed dense graphs are antimagic) — but the general connected case resists: the obvious counting obstructions vanish, yet no construction covers all graphs, and no counterexample has surfaced in over three decades.
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