The cyclic difference rainbow embedding conjecture

From papers

Let TT be a tree with nn edges. Let AA and BB be two copies of Zn\mathbb{Z}_n, and form the complete bipartite graph between them by colouring each edge (a,b)A×B(a,b)\in A\times B with baZnb-a\in\mathbb{Z}_n. A copy is rainbow if all its edge colours are distinct.

Cyclic difference rainbow embedding conjecture. The resulting edge-coloured bipartite graph contains a rainbow embedding of TT.

Such an embedding would yield the Graham–Häggkvist decomposition by cyclic translations. The paper expects bounded-degree cases to follow from its methods, but the general conjecture remains open.

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Sources & referencesView supporting material

Primary source

Alp Müyesser and Alexey Pokrovskiy, “On the Graham–Sloane harmonious labelling conjecture”, arXiv:2509.05280 (2025).

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