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...

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 ...

for a given hypergraph $h$ with chromatic number $chi(h)$ and with no edge containing only one vertex, it is shown that the minimum number $l$ for which there exists a partition (also a covering) ${e_1,e_2,ldots,e_l}$ for $e(h)$, such that the hypergraph induced by $e_i$ for each $1leq ileq l$ is $k$-colorable, is $lceil log_{k} chi(h) rceil$.

In ZF the existence of self-complementary graphs on every infinite set is equivalent to the statement that every infinite set is ’even’. We prove a generalization to k-uniform t-complementary hypergraphs. 1. Self-complementary graphs A graph G is called self-complementary if it is isomorphic to its complement G, which has the same vertices as G and exactly those edges that are not in G. First w...

Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow t-matching in an r-partite r-uniform multi-hypergraph whose edges are colored with f colors such that every color class is a matching of size t. This problem was posed by Aharoni and Berger, who asked...

We compare tractable classes of constraint satisfaction problems (CSPs). We first give a uniform presentation of the major structural CSP decomposition methods. We then introduce a new class of tractable CSPs based on the concept of hypertree decomposition recently developed in Database Theory. We introduce a framework for comparing parametric decomposition-based methods according to tractabi l...