The 1-2-3-Conjecture for Hypergraphs

A weighting of the edges of a hypergraph is called vertex-coloring if the weighted degrees of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this paper we show that such a weighting is possible from the weight set {1, 2, . . . , r + 1} for all linear hypergraphs with maximum edge size r ≥ 4 and not containing isolated edges. The number r + 1 is best possible for this statement. Further, the weight set {1, 2, 3, 4, 5} is sufficient for all hypergraphs with maximum edge size 3, as well as {1, 2, . . . , 5r − 5} for all hypergraphs with maximum edge size r ≥ 4, up to some trivial