Minimum-Weight Edge Discriminators in Hypergraphs

In this paper we introduce the notion of minimum-weight edge-discriminators in hypergraphs, and study their various properties. For a hypergraph H = (V,E ), a function λ : V → Z+∪{0} is said to be an edge-discriminator onH if∑v∈Ei λ(v) > 0, for all hyperedges Ei ∈ E , and ∑ v∈Ei λ(v) 6= ∑ v∈Ej λ(v), for every two distinct hyperedges Ei, Ej ∈ E . An optimal edge-discriminator on H, to be denoted by λH, is an edge-discriminator on H satisfying ∑v∈V λH(v) = minλ∑v∈V λ(v), where the minimum is taken over all edge-discriminators on H. We prove that any hypergraph H = (V,E ), with |E | = m, satisfies ∑v∈V λH(v) 6 m(m + 1)/2, and the equality holds if and only if the elements of E are mutually disjoint. For runiform hypergraphs H = (V,E ), it follows from earlier results on Sidon sequences that ∑ v∈V λH(v) 6 |V|r+1 + o(|V|r+1), and the bound is attained up to a constant factor by the complete r-uniform hypergraph. Finally, we show that no optimal edge-discriminator on any hypergraph H = (V,E ), with |E | = m (> 3), satisfies ∑ v∈V λH(v) = m(m+ 1)/2− 1. This shows that all integer values between m and m(m+1)/2 cannot be the weight of an optimal edge-discriminator of a hypergraph, and this raises many other interesting combinatorial questions.