Property B Research Articles

Extremal problems in hypergraph colourings

Экстремальные задачи в раскрасках гиперграфов неявно берут свое начало в теоремах Гильберта об одноцветных аффинных кубах (1892) и ван дер Вардена об одноцветных арифметических прогрессиях (1927). В дальнейшем, с появлением и развитием теории Рамсея, число задач о раскраске явно заданных гиперграфов росло. Однако систематическое изучение экстремальных задач о раскрасках гиперграфов началось с работ П. Эрдeша и А. Хайнала 60-х годов XX в. Данный обзор посвящен задачам о поиске гиперграфа с минимальным числом ребер, лежащего в некотором классе гиперграфов, их вариациям и приложениям. Центральной задачей такого типа является задача Эрдeша-Хайнала о нахождении минимального числа ребер в $n$-однородном гиперграфе с хроматическим числом не менее трех. Основная цель обзора - осветить обширные продвижения в этой области за последние несколько лет. Библиография: 168 названий.

Coloring hypergraphs with bounded cardinalities of edge intersections

The paper deals with an extremal problem concerning colorings of hypergraphs with bounded edge degrees. Consider the family of b-simple hypergraphs, in which any two edges do not share more than b common vertices. We prove that for n⩾n0(b), any n-uniform b-simple hypergraph with the maximum edge degree at most c⋅nrn−b is r-colorable, where c>0 is an absolute constant. We also establish some applications of the main result.

Complete colourings of hypergraphs

A complete c-colouring of a graph is a proper colouring in which every pair of distinct colours from [c]={1,2,…,c} appears as the colours of endvertices of some edge. We consider the following generalisation of this concept to uniform hypergraphs. A complete c-colouring for a k-uniform hypergraph H is a mapping from the vertex set of H to [c], such that (i) the colour set used on each edge has exactly k elements, and (ii) every k-element subset of [c] appears as the colour set of some edge. In this paper we exhibit some differences between complete colourings of graphs and hypergraphs. First, it is known that every graph has a complete c-colouring for some c.In contrast, we show an infinite family of hypergraphs H that do not admit a complete c-colouring for any c. We also extend this construction to λ-complete colourings (for 0<λ≤1), where condition (ii) is substituted with: at least λck different colour sets appear on edges.We establish upper and lower bounds on a maximum degree, which guarantees the existence of a complete colouring of any hypergraph. Moreover, we prove that it is NP-complete to determine if a given hypergraph has a λ-complete colouring. Next, we show that, unlike graphs, hypergraphs do not have the so-called interpolation property, i.e., we construct hypergraphs that have a complete r-colouring and a complete s-colouring, but no complete t-colouring for some t such that r<t<s.Finally, we investigate the notion of λ-complete colourings of graphs (i.e., 2-uniform hypergraphs). We show that λ-complete colourings have the same interpolation property as complete colourings. Moreover, we prove that it is NP-complete to decide whether a tree with c2 edges has a λ-complete c-colouring, which strengthens the result by Cairnie and Edwards (1997).

On Equitable Colorings of Hypergraphs

A two-coloring is said to be equitable if, on the one hand, there are no monochromatic edges (the coloring is regular) and, on the other hand, the cardinalities of color classes differ from one another by at most 1. It is proved that, for the existence of an equitable two-coloring, it suffices that the number of edges satisfy an estimate of the same order as that for a regular coloring. This result strengthens the previously known Radhakrishnan-Srinivasan theorem.

Mario Gionfriddo and mixed hypergraph coloring

We give a brief description of the explicit and implicit contribution of Mario Gionfriddo to mixed hypergraph coloring.

Proper Coloring of Geometric Hypergraphs

We study whether for a given planar family {mathcal {F}} there is an m such that any finite set of points can be 3-colored so that any member of {mathcal {F}} that contains at least m points contains two points with different colors. We conjecture that if {mathcal {F}} is a family of pseudo-disks, then such an m exists. We prove this in the special case when {mathcal {F}} is the family of all homothetic copies of a given convex polygon. We also study the problem in higher dimensions.

Equitable colorings of hypergraphs with few edges

The paper deals with an extremal problem concerning equitable colorings of uniform hypergraph. Recall that a vertex coloring of a hypergraph H is called proper if there are no monochromatic edges under this coloring. A hypergraph is said to be equitably r-colorable if there is a proper coloring with r colors such that the sizes of any two color classes differ by at most one. In the present paper we prove that if the number of edges |E(H)|≤0.01nlnnr−1rrn−1 then the hypergraph H is equitably r-colorable provided r<lnn5.

Edge‐coloring linear hypergraphs with medium‐sized edges

Motivated by the Erdos̋‐Faber‐Lovász (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We show that if the hyper‐edge sizes are bounded between i and inclusive, then there is a list edge coloring using colors. The dependence on n in the upper bound is optimal (up to the value of Ci,ϵ).

Equitable coloring of hypergraphs

A hypergraph is equitably k-colorable if its vertices can be partitioned into k sets/color classes in such a way that monochromatic edges are avoided and the number of vertices in any two color classes differs by at most one. We prove that the problem of equitable 2-coloring of hypergraphs is NP-complete even for 3-uniform hyperstars. Finally, we apply the method of dynamic programming for designing a polynomial-time algorithm to equitably k-color linear hypertrees, where k≥2 is fixed.

Panchromatic colorings of random hypergraphs

Работа посвящена изучению пороговой вероятности наличия полноцветной раскраски в $r$ цветов у случайного $k$-однородного гиперграфа в биномиальной модели $H(n,k,p)$, т.е. такой раскраски, что каждое ребро гиперграфа содержит вершины всех $r$ цветов. Показано, что данная пороговая вероятность при фиксированных $r&lt;k$ и растущем $n$ отвечает разреженному случаю, т.е. случаю $p=cn/{n\choose k}$ для положительного фиксированного $c$. Найдены ее очень близкие оценки в виде ограничений $c_1(r,k)&lt;c_2(r,k)$ значений параметра $c$, где разность $c_2(r,k)-c_1(r,k)$ экспоненциально быстро стремится к нулю при заданном $r$ и растущем $k$.

Rigid Colorings of Hypergraphs and Contiguity

We consider the problem of $q$-coloring a $k$-uniform random hypergraph, where $q,k \geq 3$, and determine the rigidity threshold. For edge densities above the rigidity threshold, we show that almo...

Counting Hypergraph Colorings in the Local Lemma Regime

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of $q$-colorings for $k$-uniform hypergraphs with maximum degree $\Delta$ if $k\ge 28$ and $q > 357 \Delta^{\frac{14}{k-14}}$. We also obtain a polynomial-time almost uniform sampler if $q>931 \Delta^{\frac{16}{k-16/3}}$. These are the first approximate counting and sampling algorithms in the regime $q\ll\Delta$ (for large $\Delta$ and $k$) without any additional assumptions. Our method is based on the recent work of Moitra (STOC, 2017). One important contribution of ours is to remove the dependency of $k$ and $\Delta$ in Moitra's approach.

Reverse mathematics and colorings of hypergraphs

Working in subsystems of second order arithmetic, we formulate several representations for hypergraphs. We then prove the equivalence of various vertex coloring theorems to $${\textsf {WKL}}_0$$ , $${\textsf {ACA}}_0$$ , and $${\varPi ^1_1 \text {-}}{\textsf {CA}}_0$$ .

Conflict-free connection of trees

We study the conflict-free connection coloring of trees, which is also the conflict-free coloring of the so-called edge-path hypergraphs of trees. We first prove that for a tree T of order n, $$ cfc (T)\ge cfc (P_n)=\lceil \log _{2} n\rceil $$ , which completely confirms the conjecture of Li and Wu [24]. We then present a sharp upper bound for the conflict-free connection number of trees by a simple algorithm. Furthermore, we show that the conflict-free connection number of the binomial tree with $$2^{k-1}$$ vertices is $$k-1$$ . At last, we study trees which are $$ cfc $$ -critical, and prove that if a tree T is $$ cfc $$ -critical, then the conflict-free connection coloring of T is equivalent to the edge ranking of T.

Polychromatic colorings and cover decompositions of hypergraphs

A polychromatic 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. A polychromatic m-coloring of a hypergraph H corresponds to a cover m-decomposition of its dual hypergraph H*. The maximum integer m that a hypergraph H admits a cover m-decomposition is exactly the longest lifetime for a wireless sensor network (WSN) corresponding to the hypergraph H. In this paper, we show that every hypergraph H has a polychromatic m-coloring if m≤⌊Sln(cΔS2)⌋, where 0 < c < 1, and Δ ≥ 1, S ≥ 2 are the maximum degree, the minimum size for all hyperedges in H, respectively. This result improves a result of Henning and Yeo on polychromatic colorings of hypergraphs in 2013, and its dual form improves one of Bollobás, Pritchard, Rothvoß, and Scott on cover decompositions of hypergraphs in 2013. Furthermore, we give a sufficient condition for a hypergraph H to have an “equitable” polychromatic coloring, which extends the result of Henning and Yeo in 2013 and improves in part one of Beck and Fiala in 1981 on 2-colorings (property B) of hypergraphs.

A new lower bound for the chromatic number of general Kneser hypergraphs

A general Kneser hypergraph KGr(H) is an r-uniform hypergraph that somehow encodes the edge intersections of a ground hypergraph H. The colorability defect of H is a combinatorial parameter providing a lower bound for the chromatic number of KGr(H), which is addressed in a series of works by Dol’nikov (1988), Kříž (1992) and Ziegler (2002). In this paper, we define a new combinatorial parameter, the equitable colorability defect of hypergraphs, which provides some common improvements upon these works. Roughly speaking, we propose a new lower bound for the chromatic number of general Kneser hypergraphs which substantially improves Ziegler’s lower bound. It is always as good as Ziegler’s lower bound and several families of hypergraphs for which the difference between these two lower bounds is arbitrary large are provided. This specializes to a substantial improvement of the Dol’nikov–Kříž lower bound for the chromatic number of general Kneser hypergraphs as well. Furthermore, we prove a result ensuring the existence of a colorful subhypergraph in any proper coloring of general Kneser hypergraphs that strengthens Meunier’s result (Meunier, 2014).

Exact and Fixed Parameter Tractable Algorithms for Max-Conflict-Free Coloring in Hypergraphs

Conflict-free coloring of hypergraphs is a very well studied question of theoretical and practical interest. For a hypergraph $H=(U, \mathcal{F})$, a conflict-free coloring of $H$ refers to a verte...

Phase transitions in the q-coloring of random hypergraphs

We study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied. Hypergraph coloring is a constraint satisfaction problem where each constraint includes K variables that must be assigned one out of q colors in such a way that there are no monochromatic constraints, i.e. there are at least two distinct colors in the set of variables belonging to every constraint. This problem generalizes naturally coloring of random graphs () and bicoloring of random hypergraphs (), both of which were extensively studied in past works. The study of random hypergraph coloring gives us access to a case where both the size q of the domain of the variables and the arity K of the constraints can be varied at will. Our work provides explicit values and predictions for a number of phase transitions that were discovered in other constraint satisfaction problems but never evaluated before in hypergraph coloring. Among other cases we revisit the hypergraph bicoloring problem () where we find that for and the colorability threshold is not given by the one-step-replica-symmetry-breaking analysis as the latter is unstable towards more levels of replica symmetry breaking. We also unveil and discuss the coexistence of two different 1RSB solutions in the case of , . Finally we present asymptotic expansions for the density of constraints at which various phase transitions occur, in the limit where q and/or K diverge.

Colourings of Uniform Hypergraphs with Large Girth and Applications

This paper deals with a combinatorial problem concerning colourings of uniform hypergraphs with large girth. We prove that ifHis ann-uniform non-r-colourable simple hypergraph then its maximum edge degree Δ(H) satisfies the inequality$$ \Delta(H)\geqslant c\cdot r^{n-1}\ffrac{n(\ln\ln n)^2}{\ln n} $$for some absolute constantc&gt; 0.As an application of our probabilistic technique we establish a lower bound for the classical van der Waerden numberW(n, r), the minimum naturalNsuch that in an arbitrary colouring of the set of integers {1,. . .,N} withrcolours there exists a monochromatic arithmetic progression of lengthn. We prove that$$ W(n,r)\geqslant c\cdot r^{n-1}\ffrac{(\ln\ln n)^2}{\ln n}. $$

Colorings of b-simple hypergraphs

The paper deals with an extremal problem concerning colorings of hypergraphs with bounded edge degrees. Consider the family of b-simple hypergraphs, in which any two edges do not share more than b common vertices. We prove that for n≥n0(b), any n-uniform b-simple hypergraph with the maximum edge degree at most cnrn−b is r-colorable, where c>0 is an absolute constant.