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On Perfect Balanced Rainbow-Free Colorings and Complete Colorings of Projective Spaces
Abstract
This paper is motivated by the problem of determining the related chromatic numbers of some hypergraphs. A hypergraph is defined from a projective space PG, where the vertices are points and the hyperedges are -dimensional subspaces. For the perfect balanced rainbow-free colorings, we show that , where and l is the smallest nontrivial factor of . For the complete colorings, we prove that there is no complete coloring for with . We also provide some results on the related chromatic numbers of subhypergraphs of .
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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.
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.
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.
KeywordsEdge-coloring-Ramsey theory-Rainbow-Heterochromatic-Multicolored-Anti-Ramsey
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.
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)(lnn)/n is the threshold function for the existence of a rainbow-free colouring in a random k-uniform hypergraph.
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.
A 3-coloring of the elements of an abelian group is said to be rainbow-free if there is no 3-term arithmetic progression with its members having pairwise distinct colors. We give a structural characterization of rainbow-free colorings of abelian groups. This characterization proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow-free 3-coloring of cyclic groups.
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).
A harmonious coloring of a k-uniform hypergraph H is a rainbow vertex coloring such that each k-set of colors appears on at most one edge. A rainbow coloring of H is achromatic if each k-set of colors appears on at least one edge. The harmonious number (resp. achromatic number) of H, denoted by h(H) (resp. ψ(H)) is the minimum (resp. maximum) possible number of colors in a harmonious (resp. achromatic) coloring of H. A class H of hypergraphs is fragmentable if for every H∈H, H can be fragmented into components of a bounded size by removing a “small” fraction of vertices.
We show that for every fragmentable class H of bounded degree hypergraphs, for every ϵ>0 and for every hypergraph H∈H with m≥m0(H,ϵ) edges we have h(H)≤(1+ϵ)k!mk and ψ(H)≥(1−ϵ)k!mk.
As corollaries, we answer a question posed by Blackburn concerning the maximum length of t-subset packing sequences of constant radius and derive an asymptotically tight bound on the minimum number of colors in a vertex-distinguishing edge coloring of cubic planar graphs, which is a step towards confirming a conjecture of Burris and Schelp.
We prove a sufficient and a necessary condition for a square-free monomial ideal J associated to a (dual) hypergraph to have projective dimension equal to the minimal number of generators of J minus 2. We also provide an effective explicit procedure to compute the projective dimension of 1-dimensional hypergraphs when each connected component contains at most one cycle. An algorithm to compute the projective dimension is also included. Applications of these results are given; they include, for instance, computing the projective dimension of monomial ideals whose associated hypergraph has a spanning Ferrers graph.
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).
Given a hypergraph and a set of colors, we want to find a vertex coloring to minimize the size of any monochromatic set in an edge. We give a deterministic polynomial time approximation algorithm with performance close to the best bound guaranteed by an existential argument. This can be applied to support divide and conquer approaches to various problems. We give two examples. For deterministic DNF approximate counting, this helps us explore the importance of a previously ignored parameter, the maximum number of appearances of any variable, and we construct algorithms that are particularly good when this parameter is small. For partially ordered sets, we are able to constructivize the dimension bound given by Furedi and Kahn [Order, 3 (1986), pp. 15 - 20].
Some problems on representations of subspaces in a finite field are discussed and a new proof about Singer difference sets is given. Finally, a class of association schemes is constructed by all affine hyperplanes in a finite field and the parameters are computed.
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.
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
We present a simple result on coloring hypergraphs and use it to obtain bounds on the chromatic number of graphs which do not induce certain trees.
A mixed hypergraph is a triple (V,C,D) where V is the vertex set and C and D are families of subsets of V called C-edges and D-edges, respectively. A proper coloring of a mixed hypergraph (V,C,D) is a coloring of its vertices such that no C-edge is polychromatic and no D-edge is monochromatic. We show that mixed hypergraphs can be used to efficiently model several graph coloring problems including homomorphisms of simple graphs and multigraphs, circular colorings, (H,C,⩽K)-colorings, (H,C,K)-colorings, locally surjective, locally bijective and locally injective homomorphisms, L(p,q)-labelings, the channel assignment problem, T-colorings and generalized T-colorings.
A theorem in finite goemetry and some applications to number theory
- J A Singer
Combinatorial Design Theory
- H Shen