Weak and Strong Versions of the 1-2-3 Conjecture for Uniform Hypergraphs

Abstract

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]

Weight choosability of oriented hypergraphs

“…. , r} such that the weighted degrees W v form a proper coloring of H. The following conjecture is stated in [4] (see also [7]). The conjecture holds for random uniform hypergraphs in a stronger sense that even non-weighted degrees give a proper coloring, as proved recently in [4].…”

On offset Hamilton cycles in random hypergraphs

“…Furthermore, they conjectured that each 3-uniform hypergraph without isolated edges is weakly 3-weighted. It was shown by Bennett, Dudek, Frieze and Helenius [4] that for almost all uniform hypergraphs these conjectures hold.…”

The 1-2-3-Conjecture for Hypergraphs

“…So let x 1 < x 2 < x 3 be such remaining choices for ρ(v n−3 v n−2 ), and let y 1 < y 2 < y 3 be such remaining choices for ρ(v n−1 v n ). 3 , so we have at least five different options for ρ(v n−1 v n ) − ρ(v n−3 v n−2 ) at this point. Therefore, at least one of these options will yield a result with ρ(v n−i ) = ρ(v n− j ) for all 0 ≤ i ≤ 1, 2 ≤ j ≤ 3.…”

“…At some point we thought that it may even be true that for r ≥ 4, the set {1, 2} is sufficient, and in fact asked this question in a previous version of this manuscript. Then Bennett et al [3] constructed a family of uniform hypergraphs for all r that require the set {1, 2, 3}, and extended Conjecture 10 accordingly.…”

“…For every r-uniform hypergraph H without an isolated edge, there is a weighting ρ : E(H ) → {1, 2, 3}, such that the induced vertex weights ρ(v) := e v ρ(e) properly color V (H ). The construction in [3] is as follows. Start with a cycle of length 4k + 2, and transform every 2-edge into an r-edge by producing (r − 2) twins of every other vertex in the cycle.…”