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The 1-2-3 conjecture for 3-uniform hypergraphs

Let H=(V,E)\mathcal{H}=(V,E) be a hypergraph, and call it 33-uniform when every edge has size 33. A hypergraph vertex coloring is proper when every edge contains at least two vertices with distinct colors. For an edge weighting w:E(H)→{1,2,…,k}w:E(\mathcal{H})\rightarrow\{1,2,\ldots,k\}, the induced vertex color of vv is the sum of the weights of edges containing vv, and χe(H)\chi^{e}(\mathcal{H}) is the least such kk. The 1-2-3 conjecture for 3-uniform hypergraphs. For every 33-uniform hypergraph H\mathcal{H} with no isolated edge, χe(H)≤3\chi^{e}(\mathcal{H})\leq 3. This generalizes the graph 1-2-3 conjecture to 3-uniform hypergraphs. The source notes bounds for general uniformity but does not report a resolution of this assertion.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The 1-2-3 Conjecture for 3-uniform hypergraphs

    Let HH be a 33-uniform hypergraph without an isolated edge. A weighting ω:E(H)→{1,2,3}\omega:E(H)\to\{1,2,3\} induces vertex weights

    ω(v):=∑e∋vω(e).\omega(v):=\sum_{e\ni v}\omega(e).

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

    This is the 33-uniform analogue of the graph 1-2-3 conjecture. The source states it as an open question and gives no resolution status.

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

Sources & referencesView supporting material

Primary source

Akbar Davoodi and Leila Maherani, “On the total versions of 1-2-3-conjecture for graphs and hypergraphs”, arXiv:2204.13936 (2022).

edge-weightingsgraph-weightingshypergraphsuniform-hypergraphsvertex-coloring

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