research

Minimum-Weight Edge Discriminator in Hypergraphs

Abstract

In this paper we introduce the concept of minimum-weight edge-discriminators in hypergraphs, and study its various properties. For a hypergraph H=(V,E)\mathcal H=(\mathcal V, \mathcal E), a function Ξ»:Vβ†’Z+βˆͺ{0}\lambda: \mathcal V\rightarrow \mathbb Z^{+}\cup\{0\} is said to be an {\it edge-discriminator} on H\mathcal H if βˆ‘v∈EiΞ»(v)>0\sum_{v\in E_i}{\lambda(v)}>0, for all hyperedges Ei∈EE_i\in \mathcal E, and βˆ‘v∈EiΞ»(v)β‰ βˆ‘v∈EjΞ»(v)\sum_{v\in E_i}{\lambda(v)}\ne \sum_{v\in E_j}{\lambda(v)}, for every two distinct hyperedges Ei,Ej∈EE_i, E_j \in \mathcal E. An {\it optimal edge-discriminator} on H\mathcal H, to be denoted by Ξ»H\lambda_\mathcal H, is an edge-discriminator on H\mathcal H satisfying βˆ‘v∈VΞ»H(v)=minβ‘Ξ»βˆ‘v∈VΞ»(v)\sum_{v\in \mathcal V}\lambda_\mathcal H (v)=\min_\lambda\sum_{v\in \mathcal V}{\lambda(v)}, where the minimum is taken over all edge-discriminators on H\mathcal H. We prove that any hypergraph H=(V,E)\mathcal H=(\mathcal V, \mathcal E), with ∣E∣=n|\mathcal E|=n, satisfies βˆ‘v∈VΞ»H(v)≀n(n+1)/2\sum_{v\in \mathcal V} \lambda_\mathcal H(v)\leq n(n+1)/2, and equality holds if and only if the elements of E\mathcal E are mutually disjoint. For rr-uniform hypergraphs H=(V,E)\mathcal H=(\mathcal V, \mathcal E), it follows from results on Sidon sequences that βˆ‘v∈VΞ»H(v)β‰€βˆ£V∣r+1+o(∣V∣r+1)\sum_{v\in \mathcal V}\lambda_{\mathcal H}(v)\leq |\mathcal V|^{r+1}+o(|\mathcal V|^{r+1}), and the bound is attained up to a constant factor by the complete rr-uniform hypergraph. Next, we construct optimal edge-discriminators for some special hypergraphs, which include paths, cycles, and complete rr-partite hypergraphs. Finally, we show that no optimal edge-discriminator on any hypergraph H=(V,E)\mathcal H=(\mathcal V, \mathcal E), with ∣E∣=n(β‰₯3)|\mathcal E|=n (\geq 3), satisfies βˆ‘v∈VΞ»H(v)=n(n+1)/2βˆ’1\sum_{v\in \mathcal V} \lambda_\mathcal H (v)=n(n+1)/2-1, which, in turn, raises many other interesting combinatorial questions.Comment: 22 pages, 5 figure

    Similar works

    Full text

    thumbnail-image

    Available Versions