Article

Upper chromatic number of finite projective planes

Authors:
Gábor Bacsó at ELKH
Gábor Bacsó
  • ELKH
Zsolt Tuza at University of Pannonia
Zsolt Tuza
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Abstract

For a finite projective plane , let denote the maximum number of classes in a partition of the point set, such that each line has at least two points in the same partition class. We prove that the best possible general estimate in terms of the order of projective planes is , which is tight apart from a multiplicative constant in the third term : Our results asymptotically solve a ten-year-old open problem in the coloring theory of mixed hypergraphs, where is termed the upper chromatic number of . Further improvements on the upper bound (1) are presented for Galois planes and their subclasses. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 221–230, 2008

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... In [1] the following general bound is given on the upper chromatic number for any projective plane, as a function of the order, and thus a ten-year-old open problem is solved in the coloring theory of mixed hypergraphs. ...
... Result 1.2 (Bacsó, Tuza [1]). As q → ∞, any projective plane Π q of order q satisfies ...
... We recall a more general result of [1]. Let τ 2 (Π q ) = 2(q +1)+c(Π q ). ...
The 2-Blocking Number and the Upper Chromatic Number of PG (2, q )
Article
Full-text available
  • Apr 2013
AbstractA twofold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ2(Π). Let PG(2,q) be the Desarguesian projective plane over GF(q), the finite field of q elements. We show that if q is odd, not a prime, and r is the order of the largest proper subfield of GF(q), then τ2PG(2,q))≤ 2(q+(q‐1)/(r‐1)). For a finite projective plane Π, let χ¯(Π) denote the maximum number of classes in a partition of the point‐set, such that each line has at least two points in some partition class. It can easily be seen that χ¯(Π)≥v−τ2(Π)+1 (⋆) for every plane Π on v points. Let q=ph, p prime. We prove that for Π= PG (2,q), equality holds in (⋆) if q and p are large enough.
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... In recent years the notion of a proper strict coloring of hypergraphs was investigated in several papers by Voloshin, Bacsó, Tuza and others, including [1], [2], [3] and [4]. In this work, instead of studying the upper chromatic number we will focus on improving the known estimates of the balanced upper chromatic number of such hypergraphs which arise from projective planes. ...
... We obtain a planar difference set by starting with a primitive cubic polynomial p(x) = x 3 −ax 2 −bx−c over GF(q) and now define the field GF(q 3 ) = GF(q)[x]/(p(x)). Every monomial x i now reduces to a degree (at most) 2 polynomial c 2 x 2 +c 1 x+c 0 ≡ (c 2 , c 1 , c 0 ) ∈ GF(q) 3 . The exponents i, with 0 ≤ i ≤ q 2 +q for which x i lies in a two-dimensional subspace now give a difference set. ...
On the balanced upper chromatic number of finite projective planes
Preprint
Full-text available
  • May 2020
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)\mathrm{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.
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... We denote this hypergraph by n k q ( , , ); however, we usually take into account the richer structure of n q PG( , ) when working in n k q ( , , ). The study of this particular case was started in the mid-nineties by Bacsó and Tuza [3], who established general bounds for the upper chromatic number of arbitrary finite projective planes (considered as a hypergraph whose points and hyperedges are the points and lines of the plane). Let us introduce the notation θ θ q q q ...
... Result 1.4 (Bacsó and Tuza [3]). Let Π q be an arbitrary finite projective plane of order q, and let τ q c (Π ) = 2( + 1) + (Π ) . ...
On the upper chromatic number and multiple blocking sets of PG(n,q)
Article
Full-text available
  • Nov 2019
We investigate the upper chromatic number of the hypergraph formed by the points and the k‐dimensional subspaces of PG(n,q); that is, the most number of colors that can be used to color the points so that every k‐subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for t≤38p+1, a small t‐fold (weighted) (n−k)‐blocking set of PG(n,p),p prime, must contain the weighted sum of t not necessarily distinct (n−k)‐spaces.
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... In recent years the notion of a proper strict coloring of hypergraphs was investigated in several papers by Voloshin, Bacsó, Tuza and others, including [3,15,16] and [2]. In this work, instead of studying the upper chromatic number we will focus on improving the known estimates of the balanced upper chromatic number of such hypergraphs which arise from projective planes. ...
On the balanced upper chromatic number of finite projective planes
Article
  • Mar 2021
  • DISCRETE MATH
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.
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... The upper chromatic number has been studied in many different contexts and has been redefined several times under different names (see [2,3,6,7,10,13,14] and references therein). More specifically, results in projective planes appear for example in [1,4,5]. A natural lower bound for the upper chromatic number is given in terms of the 2-transversal number τ 2 , which is the minimum cardinality of a set of vertices that intersect each edge in at least two vertices. ...
On the balanced upper chromatic number of cyclic projective planes and projective spaces
Article
  • Jul 2015
  • DISCRETE MATH
We study vertex colorings of hypergraphs, such that all color sizes differ at most in one (balanced colorings) and each edge contains at least two vertices of the same color (rainbow-free colorings). Given a hypergraph H, the maximum k, such that there is a balanced rainbow-free k-coloring of H is called the balanced upper chromatic number denoted by χ¯b(H). Concerning hypergraphs defined by projective spaces, bounds on the balanced upper chromatic number and constructions of rainbow-free colorings are given. For cyclic projective planes of order q we prove that: q2 + q + 1/6≤χ¯b(Πq)≤q2 + q + 1/3. We also give bounds for the balanced upper chromatic numbers of the hypergraphs arising from the n-dimensional finite space PG(n, q).
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... Concerning the decrements of projective planes Π(q) of order q, Bacsó and Tuza [4] proved that dec(Π(q)) ≥ 2q + √ q/2−o( √ q). For an infinite sequence of values q this is provably optimal, even in the order Θ( √ q) of its second term: if q is a square, then the union of two disjoint Baer subplanes in the Galois plane P G(2, q) meets each line in more than one point, and this allows us to construct a colouring; from which we deduce that dec(P G(2, q)) ≤ 2q +2 ...
Hypergraph Colouring
Chapter
Full-text available
  • Jun 2015
We discuss the colouring theory of finite set systems. This is not merely an extension of results from collections of 2-element sets (graphs) to larger sets. The wider structure (hypergraphs) offers many interesting new kinds of problems, which either have no analogues in graph theory or become trivial when we restrict them to graphs. Introduction In this introductory section we give the most important definitions required to study hypergraph colouring, and briefly survey the half-century history of this topic. For more details on the material of Sections 1 and 2 we refer to Berge [8], Zykov [76] and Duchet [27]. Let V = {v1, v2, …, vn} be a finite set of elements called vertices, and let ℇ = {E1, E2, …, Em} be a family of subsets of V called edges or hyperedges. The pair ℌ = (V, ℇ) is called a hypergraph with vertex-set V = V(ℌ) and edge-set ℇ = ℇ(ℌ). The hypergraph ℌ = (V, ℇ) is sometimes called a set system. If each edge of a hypergraph contains precisely two vertices, then it is a graph. As in graph theory, the number |V| = n is called the order of the hypergraph. Edges with fewer than two elements are usually allowed, but will be disregarded here. Thus, throughout this chapter we assume that each edge E ∈ ℇ contains at least two vertices, unless stated explicitly otherwise. Edges that coincide are called multiple edges. In a hypergraph, two vertices are said to be adjacent if there is an edge containing both of these vertices. The adjacent vertices are sometimes called neighbours of each other, and the set of neighbours of a given vertex v is called the (open) neighbourhood N(v) of v. If v ∈ E, then the vertex v and the edge E are incident with each other. For an edge E, the number |E| is called the size or cardinality of E.
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Maximum number of colors in hypertrees of bounded degree
Article
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  • Jan 2014
The upper chromatic number χ‾(H)\overline{\chi }(\mathcal{H}) of a hypergraph H=(X,E)\mathcal{H}=(X,\mathcal{E}) is the maximum number of colors that can occur in a vertex coloring φ:X→N\varphi :X\rightarrow \mathbb {N} such that no edge E∈EE\in \mathcal{E} is completely multicolored. A hypertree (also called arboreal hypergraph) is a hypergraph whose edges induce subtrees on a fixed tree graph. It has been shown that on hypertrees it is algorithmically hard not only to determine exactly but also to approximate the value of χ‾\overline{\chi } , unless P=NP\mathsf{P}=\mathsf{NP} . In sharp contrast to this, here we prove that if the input is restricted to hypertrees H\mathcal{H} of bounded maximum vertex degree, then χ‾(H)\overline{\chi }(\mathcal{H}) can be determined in linear time if an underlying tree is also given in the input. Consequently, χ‾\overline{\chi } on hypertrees is fixed parameter tractable in terms of maximum degree.
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... A C-coloring ϕ with |ϕ(X)| = χ(H) colors will be referred to as an optimal coloring of H. The decrement of H = (X, E), introduced in [2], is defined as dec(H) = n − χ(H). Similarly, the decrement of a C-coloring ϕ : X → N is meant as dec(ϕ) = |X| − |ϕ(X)|. ...
Approximability of the upper chromatic number of hypergraphs
Article
Full-text available
  • Oct 2013
  • DISCRETE MATH
A C-coloring of a hypergraph H=(X,E){\cal H}=(X,{\cal E}) is a vertex coloring φ:X→N\varphi: X\to {\mathbb{N}} such that each edge E∈EE\in{\cal E} has at least two vertices with a common color. The related parameter χ‾(H)\overline{\chi}({\cal H}), called the upper chromatic number of H{\cal H}, is the maximum number of colors can be used in a C-coloring of H{\cal H}. A hypertree is a hypergraph which has a host tree T such that each edge E∈EE\in {\cal E} induces a connected subgraph in T. Notations n and m stand for the number of vertices and edges, respectively, in a generic input hypergraph. We establish guaranteed polynomial-time approximation ratios for the difference n−χ‾(H)n-\overline{\chi}({\cal H}), which is 2+2ln⁡(2m)2+2 \ln (2m) on hypergraphs in general, and 1+ln⁡m1+ \ln m on hypertrees. The latter ratio is essentially tight as we show that n−χ‾(H)n-\overline{\chi}({\cal H}) cannot be approximated within (1−ϵ)ln⁡m(1-\epsilon) \ln m on hypertrees (unless NP⊆DTIME(nO(log  log  n)){\sf NP} \subseteq {\sf DTIME} (n^{{\cal O}(log\;log\; n)})). Furthermore, χ‾(H)\overline{\chi}({\cal H}) does not have O(n1−ϵ){\cal O}(n^{1-\epsilon})-approximation and cannot be approximated within additive error o(n) on the class of hypertrees (unless P=NP{\sf P}={\sf NP}).
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... This yields χ ≥ q 2 −2q +3 in general. The strongest general upper bound, proved in [3], is somewhat larger in its second term and has the form ...
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  • Aug 2011
We discuss problems and results on the maximum number of colors in combinatorial structures under the assumption that no totally multicolored sets of a specified type occur.
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... It was an open problem since the mid-1990's whether or not 3q is a nearly tight estimate. Recently, it has been proved by Bacsó and Tuza [1] that ...
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Invited talk at the International Conference Combinatorics 2012: 1912-2012: �One hundred years to Chromatic Polynomial. Perugia, Italy, 2012.
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... A survey on colorings of mixed Steiner systems can be found in [38]. Very recently, it has been proved by Bacsó and Tuza in [1] that the best possible general upper bound for the upper chromatic number of finite projective planes of order q is equal to q 2 − q − Θ √ q as q tends to infinity, both when considered as C-and bi-hypergraphs. ...
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  • Jul 2013
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... A survey on colorings of mixed Steiner systems can be found in [38]. Very recently, it has been proved by Bacsó and Tuza in [1] that the best possible general upper bound for the upper chromatic number of finite projective planes of order q is equal to q 2 − q − Θ √ q as q tends to infinity, both when considered as C-and bi-hypergraphs. ...
Problems and Results on Colorings of Mixed Hypergraphs
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We survey results and open problems on ‘mixed hypergraphs’ that are hypergraphs with two types of edges. In a proper vertex coloring the edges of the first type must not be monochromatic, while the edges of the second type must not be completely multicolored. Though the first condition just means ‘classical’ hypergraph coloring, its combination with the second one causes rather unusual behavior. For instance, hypergraphs occur that are uncolorable, or that admit colorings with certain numbers k′ and k″ of colors but no colorings with exactly k colors for any k′ < k < k″.
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This paper is motivated by the problem of determining the related chromatic numbers of some hypergraphs. A hypergraph Πq(n,k)\Pi _{q}(n,k) is defined from a projective space PG(n−1,q)(n-1,q), where the vertices are points and the hyperedges are (k−1)(k-1)-dimensional subspaces. For the perfect balanced rainbow-free colorings, we show that χ‾p(Πq(n,k))=qn−1l(q−1){\overline{\chi }}_{p}(\Pi _{q}(n,k))=\frac{q^n-1}{l(q-1)}, where k≥⌈n+12⌉k\ge \lceil \frac{n+1}{2}\rceil and l is the smallest nontrivial factor of qn−1q−1\frac{q^n-1}{q-1}. For the complete colorings, we prove that there is no complete coloring for Πq(n,k)\Pi _{q}(n,k) with 2≤k<n2\le k<n. We also provide some results on the related chromatic numbers of subhypergraphs of Πq(n,k)\Pi _{q}(n,k).
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  • Sep 2019
We investigate the upper chromatic number of the hypergraph formed by the points and the k-dimensional subspaces of PG(n,q)\mathrm{PG}(n,q); that is, the most number of colors that can be used to color the points so that every k-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for t≤38p+1t\leq \frac38p+1, a small t-fold (weighted) (n−k)(n-k)-blocking set of PG(n,p)\mathrm{PG}(n,p), p prime, must contain the weighted sum of t not necessarily distinct (n−k)(n-k)-spaces.
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We introduce the notion of a co-edge of a hypergraph, which is a subset of vertices to be colored so that at least two vertices are of the same color. Hypergraphs with both edges and co-edges are called mixed hypergraphs. The maximal number of colors for which there exists a mixed hypergraph coloring using all the colors is called the upper chromatic number of a hypergraph H and is denoted by (H). An algorithm for computing the number of colorings of a mixed hypergraph is proposed. The properties of the upper chromatic number and the colorings of some classes of hypergraphs are discussed. A greedy polynomial time algorithm for finding a lower bound for (H) of a hypergraph H containing only co-edges is presented.
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Coloring Mixed Hypergraphs: Theory, Algorithms and Applications
Book
  • Jan 2002
AMS Review: This is a very interesting and magnificent book on colourings. It belongs on the shelves of everyone who works not only in graph and hypergraph theory, but more generally in discrete mathematics. The new idea of the author, to study a type of colourings different from the classic definition and all its generalizations, is described. The main feature is that mixed hypergraphs represent structures in which problems on both the minimum and maximum number of colours occur. The author develops the theory with all the results obtained to date. This book will be a useful reference text for people who study hypergraphs as well as related fields and applications. The level of the text is aimed at graduate and research use. In the introduction, the author gives an overview of graph colouring, introduces the idea of mixed hypergraph colouring, and describes unforeseen features and philosophical motivation. In the first chapter he surveys results related to the lower chromatic number. Subsequent chapters are devoted to uncolourable (having no colourings), uniquely colourable (having a unique feasible partition), C-perfect (having perfection with respect to the upper chromatic number), interval (generalizations of interval hypergraphs), pseudo-chordal (generalizations of chordal graphs) and circular (generalizations of cycle) mixed hypergraphs. Of special interest and fundamental importance are the chapters describing the gaps in the chromatic spectrum (they are not possible in classic colourings), planar mixed hypergraphs (generalizations of planar graphs) and colourings of block designs (Steiner triple and quadruple systems), considered as mixed hypergraphs. The last chapter contains 10 models of application of the concept of mixed hypergraph (ranging from computer science to molecular biology). Each chapter ends with a list of open problems, and the book contains many algorithms. It is worth mentioning that the author maintains the mixed hypergraph colouring web site at http://math.net.md/voloshin/mh.html, which in addition to detailed material about the monograph contains a list of all publications on this new scientific direction.
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New lower bounds are given for the size of a point set in a Desarguesian projective plane over, a finite field that contains at least a prescribed number s of points on every line. These bounds are best possible when q is square and s is small compared with q. In this case the smallest set is shown to be the union of disjoint Baer subplanes. The results are based on new results on the structure of certain lacunary polynomials, which can be regarded as a generalization of Rédei's results in the case when the derivative of the polynomial vanishes.
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A mixed hypergraph consists of the vertex set X and two families of subsets: the family of edges and the family of co-edges. In a coloring every edge has at least two vertices of different colors, while every co-edge has at least two vertices of the same color. The largest (smallest) number of colors for which there exists a coloring of a mixed hypergraph using all the colors is called the upper (lower) chromatic number and is denoted . A mixed hypergraph is called uncolorable if it admits no coloring. We show that there exist uncolorable mixed hypergraphs with arbitrary difference between the upper chromatic number of and the lower chromatic number of Moreover, for any , the minimum number v(k) of vertices of an inclusionwise minimal uncolorable mixed hypergraph is exactly k+4. We introduce a measure of uncolorability (the vertex uncolorability number) and propose a greedy algorithm that finds an estimate on it. We also show that the colorability problem can be expressed in terms of integer programming. Concerning particular cases, we describe those complete (l,m)-uniform mixed hypergraphs which are uncolorable, and observe that for any fixed (l,m) almost all complete (l,m)-uniform mixed hypergraphs are uncolorable, whereas generally almost all complete mixed hypergraphs are colorable.
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The upper chromatic number <(chi)over bar>(H) of a set system H is the maximum number of colours that can be assigned to the elements of the underlying set of H in such a way that each H is an element of H contains a monochromatic pair of elements. We prove that a Steiner triple system of order upsilon less than or equal to 2(k)-1 has an upper chromatic number which is at most k. This bound is the best possible, and the extremal configurations attaining equality can be characterized. Some consequences for Steiner quadruple systems are also obtained.
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  • Jan 1956
  • 975
  • Hall
The mixed hypergraphs
  • Jan 1993
  • 45
  • Voloshin V.