Property B Research Articles

On Some Generalizations of the Property B Problem of an n-Uniform Hypergraph

The extremal problem of hypergraph colorings related to the Erdős–Hajnal property B-problem is considered. Let k be a natural number. The problem is to find the value of mk(n) equal to the minimal number of edges in an n-uniform hypergraph that does not admit 2-colorings of the vertex set such that every edge of the hypergraph contains at least k vertices of each color. In this paper, we obtain new lower bounds for mk(n).

Noncontextual coloring of orthogonality hypergraphs

We discuss representations and colorings of orthogonality hypergraphs in terms of their two-valued states interpretable as classical truth assignments. Such hypergraphs, if they allow for a faithful orthogonal representation, have quantum mechanical realizations in terms of intertwined contexts or maximal observables that are widely discussed as empirically testable criteria for contextuality. A reconstruction is possible for the class of perfectly separable hypergraphs. Colorings can be constructed from a minimal set of two-valued states. Some examples from exempt categories are presented, which either cannot be reconstructed by two-valued states or whose two-valued states cannot yield a chromatic number that is equal to the maximal clique number.

Graph Entropy Based on Strong Coloring of Uniform Hypergraphs

The classical graph entropy based on the vertex coloring proposed by Mowshowitz depends on a graph. In fact, a hypergraph, as a generalization of a graph, can express complex and high-order relations such that it is often used to model complex systems. Being different from the classical graph entropy, we extend this concept to a hypergraph. Then, we define the graph entropy based on the vertex strong coloring of a hypergraph. Moreover, some tightly upper and lower bounds of such graph entropies as well as the corresponding extremal hypergraphs are obtained.

Bounds on the average and minimum attendance in preference-based activity scheduling

In this paper we study the preference-based activity scheduling problem, where based on preference sets of participants, activities need to be scheduled in parallel, such that participants can attend as many of their preferred activities as possible. The Activity Scheduling Problem (ASP) takes as input the activities, time slots, participants and their set of preferences, and is supposed to output an optimal schedule. Attendance is the number of preferred activities a participant can attend. We present two lower bounds on the average attendance over all participants, based on linearity of expectation. The first is a bound that involves the different parameters of ASP; we prove that this bound is tight. The second bound is such that no matter how many activities, participants, and time slots are included in the event, there is always a schedule that guarantees an average attendance of (1−1/e) of the preferred activities for the participants. A further result, based on applying the Lovász local lemma to hypergraph coloring, presents a lower bound on the minimum attendance over all participants. We show that for various values of the problem’s parameters this bound is better than a simple bound, although the simple bound may be tight for instances with a very large number of participants.

On rainbow-free colourings of uniform hypergraphs

We study rainbow-free colourings of k-uniform hypergraphs; that is, colourings that use k colours but with the property that no hyperedge attains all colours. We show that p⁎=(k−1)(ln⁡n)/n is the threshold function for the existence of a rainbow-free colouring in a random k-uniform hypergraph.

Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs

In colorings of some block designs, the vertices of blocks can be thought of as hyperedges of a hypergraph H that can be placed on a circle and colored according to some rules that are related to colorings of circular mixed hypergraphs. A mixed hypergraph H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or D-) induces a connected subgraph of this cycle. We propose an algorithm to color the (r,r)-uniform, complete, circular, mixed hypergraphs for all feasible values with no gaps. In doing so, we show χ(H)=2 and χ¯(H)=n−s or n−s−1 where s is the sieve number.

Cooperative Caching for Ultra-Dense Fog-RANs: Information Optimality and Hypergraph Coloring

This work considers cache placement for ultra-dense fog radio access networks (F-RANs). In an F-RAN, the fog access points (F-APs) form overlapping clusters based on their geographical locations. A cluster of F-APs then acts as a distributed cache to cooperatively serve user requests. The fronthaul traffic minimization problem is formulated in information-theoretic terms. For the k-association networks, cache placement can be optimized by concatenating an MDS code with a repetition code. By repeating the same packet in some F-APs, multicasting over the fronthaul link can be done in cache placement, which saves energy and bandwidth. Such an idea is applied to both uncoded and coded caching schemes based on hypergraph coloring. Their associated optimization problems are shown to be NP-complete. For the uncoded case, a suboptimal algorithm is proposed, which carefully repeats the subfiles. For the coded case, a heuristic algorithm that minimizes the field size requirement of the MDS repetition scheme is proposed. Simulation results demonstrate the outstanding performance of the proposed uncoded and coded caching schemes during both the cache placement phase and content delivery phase in reducing fronthaul traffic load and energy consumption.

Panchromatic colorings of random hypergraphs

Abstract We study the threshold probability for the existence of a panchromatic coloring with r colors for a random k-homogeneous hypergraph in the binomial model H(n, k, p), that is, a coloring such that each edge of the hypergraph contains the vertices of all r colors. It is shown that this threshold probability for fixed r < k and increasing n corresponds to the sparse case, i. e. the case p = c n / ( n k ) $p = cn/{n \choose k}$ for positive fixed c. Estimates of the form c 1(r, k) < c < c 2(r, k) for the parameter c are found such that the difference c 2(r, k) − c 1(r, k) converges exponentially fast to zero if r is fixed and k tends to infinity.

Generalized hypergraph coloring

A smooth hypergraph property $\mathcal{P}$ is a class of hypergraphs that is hereditary and non-trivial, i.e., closed under induced subhypergraphs and it contains a non-empty hypergraph but not all hypergraphs. In this paper we examine $\mathcal{P}$-colorings of hypergraphs with smooth hypergraph properties $\mathcal{P}$. A $\mathcal{P}$-coloring of a hypergraph $H$ with color set $C$ is a function $\varphi:V(H) \to C$ such that $H[\varphi^{-1}(c)]$ belongs to $\mathcal{P}$ for all $c \in C$. Let $L: V(H) \to 2^C$ be a so called list-assignment of the hypergraph $H$. Then, a $(\mathcal{P},L)$-coloring of $H$ is a $\mathcal{P}$-coloring $\varphi$ of $H$ such that $\varphi(v) \in L(v)$ for all $v \in V(H)$. The aim of this paper is a characterization of $(\mathcal{P},L)$-critical hypergraphs. Those are hypergraphs $H$ such $H-v$ is $(\mathcal{P},L)$-colorable for all $v \in V(H)$ but $H$ itself is not. Our main theorem is a Gallai-type result for critical hypergraphs, which implies a Brooks-type result for $(\mathcal{P},L)$-colorable hypergraphs. In the last section, we prove a Gallai bound for the degree sum of $(\mathcal{P},L)$-critical locally linear hypergraphs.

On the balanced upper chromatic number of finite projective planes

In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph H, the maximum number k for which there is a balanced rainbow-free k-coloring of H is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of Araujo-Pardo, Kiss and Montejano by determining the balanced upper chromatic number of the desarguesian projective plane PG(2,q) for all q. In addition, we determine asymptotically the balanced upper chromatic number of several families of non-desarguesian projective planes and also provide a general lower bound for arbitrary projective planes using probabilistic methods which determines the parameter up to a multiplicative constant.

On Colorings of 3-Homogeneous Hypergraphs in 3 Colors

In this paper, we examine the value m(n, r) in the Erdős–Hajnal problem. Using various methods, we obtain the estimate 27 ≤ m(3) ≤ 35.

Concepts on Coloring of Cluster Hypergraphs with Application

Coloring of graph theory is widely used in different fields like the map coloring, traffic light problems, etc. Hypergraphs are an extension of graph theory where edges contain single or multiple vertices. This study analyzes cluster hypergraphs where cluster vertices too contain simple vertices. Coloring of cluster networks where composite/cluster vertices exist is done using the concept of coloring of cluster hypergraphs. Proper coloring and strong coloring of cluster hypergraphs have been defined. Along with these, local coloring in cluster hypergraphs is also provided. Such a cluster network, COVID19 affected network, is assumed and colored to visualize the affected regions properly.

Two-Coloring Triples such that in Each Color Class Every Element is Missed at Least Once

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 linear (in the number of triples) time algorithm to decide if such a coloring exists and find one if it does. We also consider generalizations of this result and an application to a matching problem, which motivated this study. Finally, we show how these results translate to results about colorings of hypergraphs in which the degree of every vertex is k less than the number of hyperedges.

Constructing Planar Support for Non-Piercing Regions

Given a hypergraph $$\mathcal {H}=(X,{\mathcal {S}})$$, a planar support for $$\mathcal {H}$$ is a planar graph G with vertex set X, such that for each hyperedge $$S\in \mathcal {S}$$, the subgraph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph visualization. The main result proved in this paper is the following: given two families of regions R and B in the plane, each of which consists of connected, non-piercing regions, the intersection hypergraph$${\mathcal {H}}_R(B)=(B,\{B_{r}\}_{r\in R})$$, where $$B_r=\{b\in B:b\cap r\ne \emptyset \}$$ has a planar support. Further, such a planar support can be computed in time polynomial in |R|, |B|, and the number of vertices in the arrangement of the regions in $$R\cup B$$. Special cases of this result include the setting where either the family R, or the family B is a set of points. Our result unifies and generalizes several previous results on planar supports, PTAS’s for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions.

UAV-Assisted Emergency Communications in Social IoT: A Dynamic Hypergraph Coloring Approach

In this article, we address the social-awareness property and unmanned-aerial-vehicle (UAV)-assisted information diffusion in emergency scenarios, where UAVs can disseminate alert messages to a set of terrestrial users within their coverage, and then these users can continuously disseminate the received data packets to their socially connected users in a device-to-device (D2D) multicast manner. In this regard, we have to solve both the dynamic cluster formation and spectrum sharing problems in stochastic environments, since both UAVs and terrestrial users may arrive or depart suddenly. For the cluster formation problem, considering that the data rate of a multicast cluster is determined by the member with the worst link condition, we formulate it as a many-to-one matching game and adopt the rotation-swap algorithm to maximize the expected number of users receiving the alerting messages in each time slot. For the dynamic spectrum sharing problem, aiming at eliminating the interference while minimizing the channel switching cost, we propose a dynamic hypergraph coloring approach to model the cumulative interference and maintain the mutual interference at a low level by exploring a small number of vertices, when the graph is dynamically updated, i.e., the insertion/deletion of vertex/edge. Moreover, we prove some crucial properties, including global stability, convergence, and complexity. Finally, simulation results show that our proposed approach can achieve a better tradeoff among the information diffusion speed, channel switch cost, and complexity.

The List-Chromatic Number of Complete Multipartite Hypergraphs and Multiple Covers by Independent Sets

This paper deals with list colorings of uniform hypergraphs. Let H(m, r, k) be the complete r-partite k-uniform hypergraph with parts of equal size m in which each edge contains exactly one vertex from some k ≤ r parts. Using results on multiple covers by independent sets, we establish that, for fixed k and r, the list-chromatic number of H(m, r, k) is (1 + o(1)) logr/(r−k+1)(m) as m → ∞.

Colorings versus list colorings of uniform hypergraphs

AbstractLet be an integer with and be a connected ‐uniform hypergraph with edges. By refining the broken cycle theorem for hypergraphs, we show that if , then the ‐list assignment of admitting the fewest colorings is the constant list assignment. This extends the previous results of Donner, Thomassen, and the current authors for graphs.

On the connectivity of proper colorings of random graphs and hypergraphs

Let Ωq=Ωq(H) denote the set of proper [q]‐colorings of the hypergraph H. Let Γq be the graph with vertex set Ωq where two colorings σ,τ are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=Hn,m;k, k ≥ 2, the random k‐uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Γq is connected if d is sufficiently large and . This is optimal up to the first order in d. Furthermore, with a few more colors, we find that the diameter of Γq is O(n) w.h.p., where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber dynamics Markov Chain on Ωq is ergodic w.h.p.

Extremal problems in hypergraph colourings

Extremal problems in hypergraph colouring originate implicitly from Hilbert’s theorem on monochromatic affine cubes (1892) and van der Waerden’s theorem on monochromatic arithmetic progressions (1927). Later, with the advent and elaboration of Ramsey theory, the variety of problems related to colouring of explicitly specified hypergraphs widened rapidly. However, a systematic study of extremal problems on hypergraph colouring was initiated only in the works of Erdős and Hajnal in the 1960s. This paper is devoted to problems of finding edge-minimum hypergraphs belonging to particular classes of hypergraphs, variations of these problems, and their applications. The central problem of this kind is the Erdős–Hajnal problem of finding the minimum number of edges in an -uniform hypergraph with chromatic number at least three. The main purpose of this survey is to spotlight the progress in this area over the last several years. Bibliography: 168 titles.

Polychromatic colorings of hypergraphs with high balance

Let $m$ be a positive integer and $C = \{1, 2, \dots, m\}$ be a set of $m$ colors. A polychromatic $m$-coloring of a hypergraph is a coloring of its vertices in such a way that every hyperedge contains at least one vertex of each color in $C$. This problem is a generalization of $2$-colorings of hypergraphs and has close relations with the longest lifetime problem for a wireless sensor network, cover decompositions problem of hypergraphs and vertex cover problem of hypergraphs. In this paper, a main work is to find the maximum $m$ that a hypergraph $H$, with $n$ hyperedges, admits a polychromatic $m$-coloring such that each color appears at least $k$ times on each hyperedge. A $2\ln n$ approximation to the number is obtained when $k$ is a fixed positive integer. For the case that $k = O(n\ln n)$, there exists an $O(\ln n)$ approximation algorithm; for the case that $k = \omega (n\ln n)$, there exists a $(2+\sqrt{3})k$ approximation algorithm.