Given an r-uniform hypergraph H = (V, E) and a weight function ω : E → {1, . . . , w}, a coloring of vertices of H, induced by ω, is defined by c(v) = e v w(e) for all v ∈ V . If there exists such a coloring that is strong (that means in each edge no color appears more than once), then we say that H is strongly wweighted. Similarly, if the coloring is weak (that means there is no monochromatic edge), then we say that H is weakly w-weighted. In this paper, we show that almost all 3 or 4-uniform hypergraphs are strongly 2-weighted (but not 1-weighted) and almost all 5-uniform hypergraphs are either 1 or 2 strongly weighted (with a nontrivial distribution). Furthermore, for r ≥ 6 we show that almost all r-uniform hypergraphs are strongly 1-weighted. We complement these results by showing that almost all 3uniform hypergraphs are weakly 2-weighted but not 1-weighted and for r ≥ 4 almost all r-uniform hypergraphs are weakly 1-weighted. These results extend a previous work of Addario-Berry, Dalal and Reed for graphs. We also prove general lower bounds and show that there are r-uniform hypergraphs which are not strongly (r 2 − r)-weighted and not weakly 2-weighted. Finally, we show that determining whether a particular uniform hypergraph is strongly 2-weighted is NP-complete. arXiv:1511.04569v2 [math.CO]

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Abstract. An ℓ-offset Hamilton cycle C in a k-uniform hypergraph H on n vertices is a collection of edges of H such that for some cyclic order of [n] every pair of consecutive edges E i−1 , E i in C (in the natural ordering of the edges) satisfies |E i−1 ∩ E i |= ℓ and every pair of consecutive edges E i , E i+1 in C satisfies |E i ∩ E i+1 |= k − ℓ. We show that in general e k ℓ! (k − ℓ)! /n k is the sharp threshold for the existence of the ℓ-offset Hamilton cycle in the random k-uniform hypergraph H (k) n,p . We also examine this structure's natural connection to the 1-2-3 Conjecture.

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