The hypergraph 1-2-3 Conjecture

Let HH be a hypergraph whose edges have orders between 22 and rr, with no edge consisting of twins. A weighting ω:E(H){1,2,,r+1}\omega:E(H)\to\{1,2,\ldots,r+1\} induces vertex weights

ω(v):=evω(e).\omega(v):=\sum_{e\ni v}\omega(e).

Hypergraph 1-2-3 Conjecture. There is such a weighting for which the induced vertex weights properly color V(H)V(H).

The conjecture proposes that the bound proved for linear hypergraphs extends to all hypergraphs under the stated exclusion of edges consisting of twins. The source gives no resolution status.

Sources & referencesView supporting material

Primary source

Maciej Kalkowski, Michał Karoński and Florian Pfender, “The 1-2-3 Conjecture for Hypergraphs”, arXiv:1308.0611 (2016).

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