Minimum-Weight Edge Discriminator in Hypergraphs

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

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 hypergraphs with maximum edge size r>3 and not containing edges solely consisting of identical vertices. 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, up to some trivial exceptions.Comment: 12 page

Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio