The weak 3-weighting conjecture for uniform hypergraphs

From papers

Let H=(V,E)H=(V,E) be an rr-uniform hypergraph without isolated edges, where r3r\geq 3. A weight function is a map ω:E{1,2,3}\omega:E\to\{1,2,3\}, inducing the vertex-coloring c:VNc:V\to\mathbb{N} by

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

The coloring is weak if no edge is monochromatic, and HH is weakly 3-weighted if some such weight function induces a weak coloring.

Weak 3-weighting conjecture. For each r3r\geq 3, every rr-uniform hypergraph without isolated edges is weakly 3-weighted.

This extends the graph 1-2-3 conjecture to uniform hypergraphs and strengthens the previously mentioned weak weighting results. The claim is presented as an open belief in the source, with no resolution supplied.

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Sources & referencesView supporting material

Primary source

Patrick Bennett, Andrzej Dudek, Alan Frieze and Laars Helenius, “Weak and strong versions of the 1-2-3 conjecture for uniform hypergraphs”, arXiv:1511.04569 (2015).

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