Related papers: Noncontextual coloring of orthogonality hypergraph…
Perfect colorings (equitable partitions) of graphs are extensively studied, while the same concept for hypergraphs attracts much less attention. The aim of this paper is to develop basic notions and properties of perfect colorings for…
A '(partial) conflict-free coloring' of a hypergraph $\mathcal{H}$ is an assignment of colors to (a subset of) the vertex set of $\mathcal{H}$ such that every hyperedge in $\mathcal{H}$ has a vertex whose color is distinct from every other…
A 2-coloring of a hypergraph is a mapping from its vertices to a set of two colors such that no edge is monochromatic. Let $H_k(n,m)$ be a random $k$-uniform hypergraph on $n$ vertices formed by picking $m$ edges uniformly, independently…
A hypergraph is said to be $\chi$-colorable if its vertices can be colored with $\chi$ colors so that no hyperedge is monochromatic. $2$-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in…
Hypergraph states are multi-qubit states that form a subset of the locally maximally entangleable states and a generalization of the well--established notion of graph states. Mathematically, they can conveniently be described by a…
Quantum contextual sets have been recognized as resources for universal quantum computation, quantum steering and quantum communication. Therefore, we focus on engineering the sets that support those resources and on determining their…
Graph states are a fundamental class of multipartite entangled quantum states with wide-ranging applications in quantum information and computation. In this work, we develop a systematic framework for constructing and analyzing…
We give a characterization of finite sets of triples of elements (e.g., positive integers) that can be colored with two colors such that for every element $i$ in each color class there exists a triple which does not contain $i$. We give a…
In order to make more complex number-based strings from topological coding for defending against the intelligent attacks equipped with quantum computing and providing effective protection technology for the age of quantum computing, we will…
Let $\mathcal{H}$ be a hypergraph of maximal vertex degree $\Delta$, such that each its hyperedge contains at least $\delta$ vertices. Let $k=\lceil\frac{2\Delta}{\delta}\rceil$. We prove that (i) The hypergraph $\mathcal{H}$ admits proper…
Mossel and Ross raised the question of when a random colouring of a graph can be reconstructed from local information, namely the colourings (with multiplicity) of balls of given radius. In this paper, we are concerned with random…
We examine maximum vertex coloring of random geometric graphs, in an arbitrary but fixed dimension, with a constant number of colors. Since this problem is neither scale-invariant nor smooth, the usual methodology to obtain limit laws…
We study the problem of counting $k$-hypergraphlets, an interesting but surprisingly ignored primitive, with the aim of understanding whether efficient algorithms exist. To this end, we consider color coding, a well-known technique for…
One of the interesting topics in quantum contextuality is the construction for various non-contextual inequalities. By introducing a new structure called hyper-graph, we present a general method, which seems to be analytic and extensible,…
We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two $n$-colorings of a graph are said to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other…
We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the…
Compact quantum groups can be studied by investigating their co-representation categories in analogy to the Schur-Weyl/Tannaka-Krein approach. For the special class of (unitary) "easy" quantum groups these categories arise from a…
The contextuality and noncontextuality notions are considered in framework of probability representation of quantum states. Example of qutrit states and violation of the noncontextuality inequalities are presented by using the spin tomogram…
We investigate proper $(a:b)$-fractional colorings of $n$-uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned $b$ distinct colors from a set of $a$ colors, and an edge is properly colored…
A colouring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number,…