Learning a hidden hypergraph is a natural generalization of the classical group testing problem that consists in detecting unknown hypergraph Hun = H(V, E) by carrying out edge-detecting tests. In the given paper we focus our attention only on a specific family F(t, s, `) of localized hypergraphs for which the total number of vertices |V | = t, the number of edges |E| 6 s, s ¿ t, and the cardin...

Families of expander graphs are sparse graphs such that the number of vertices in each graph grows yet each graph remains difficult to disconnect. Expander graphs are of great importance in theoretical computer science. In this paper we study the connection between the Cheeger constant, a measure of the connectivity of the graph, and the smallest nonzero eigenvalue of the graph Laplacian. We sh...

For a pair of integers 1Fgr, the g-chromatic number of an r-uniform Ž . hypergraph Hs V, E is the minimal k, for which there exists a partition of V into subsets < < T , . . . , T such that elT Fg for every egE. In this paper we determine the asymptotic 1 k i Ž . behavior of the g-chromatic number of the random r-uniform hypergraph H n, p for all r Ž yrq1. possible values of g and for all value...

Suppose that T is an acyclic r-uniform hypergraph, with r ≥ 2. We define the (t-color) chromatic Ramsey number χ(T, t) as the smallest m with the following property: if the edges of any m-chromatic r-uniform hypergraph are colored with t colors in any manner, there is a monochromatic copy of T . We observe that χ(T, t) is well defined and ⌈ R(T, t)− 1 r − 1 ⌉ + 1 ≤ χ(T, t) ≤ |E(T )| + 1 where R...

We introduce the peak normal form for elements of the BaumslagSolitar groups BS(p, q). This normal form is very close to the lengthlexicographical normal form, but more symmetric. Both normal forms are geodesic. This means the normal form of an element uv yields the shortest path between u and v in the Cayley graph. The main result of this paper is that we can compute the peak normal form in po...

Let G be a finite group, and let 1G 6∈ S ⊆ G. A Cayley di-graph Γ = Cay(G,S) of G relative to S is a di-graph with a vertex set G such that, for x, y ∈ G, the pair (x, y) is an arc if and only if yx−1 ∈ S. Further, if S = S−1 := {s−1|s ∈ S}, then Γ is undirected. Γ is conected if and only if G = 〈s〉. A Cayley (di)graph Γ = Cay(G,S) is called normal if the right regular representation of G is a ...

Recently, graph neural networks have attracted great attention and achieved prominent performance in various research fields. Most of those algorithms assumed pairwise relationships objects interest. However, many real applications, the between are higher-order, beyond a formulation. To efficiently learn deep embeddings on high-order graph-structured data, we introduce two end-to-end trainable ...

The hypergraph assignment problem (HAP) is the generalization of assignments from directed graphs to directed hypergraphs. It serves, in particular, as a universal tool to model several train composition rules in vehicle rotation planning for long distance passenger railways. We prove that even for problems with a small hyperarc size and hypergraphs with a special partitioned structure the HAP ...