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On the balanced upper chromatic number of cyclic projective planes and projective spaces
Abstract
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).
... We start with a proof [1] of the fact mentioned above that ...
... By a result S.D. Cohen [7] we know that a primitive polynomial with this property exists for all q ̸ = 4. As a consequence, by means of the argument above, we proved Conjecture 1.2 since the case q = 4 has already been covered in [1]. ...
... We recall the proof of Theorem 2.3. in [1]. For 0 ≤ i ≤ q 2 +q+1 3 − 1 define the color classes as C i = ...
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.
... Result 1.1 ( [5]). All balanced rainbow-free colorings of any projective plane of order q satisfy that each color class contains at least three points. ...
... Result 1.2 ( [5]). For every cyclic projective plane Π q we have ...
... We start with a proof [5] of the fact mentioned above that χ b (Π q ) = q 2 +q+1 3 if Π q comes from a difference set containing {0, 1, 3}. ...
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 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.
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 .
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.
We distribute the points and lines of PG(2, 2n+1) according to a special structure that we call the daisy structure. This distribution is intimately related to a special block design which turns out to be isomorphic to PG(n, 2).
We show a blocking set of 3q points in PG(2, 2n+1)that intersects each line in at least two points and we apply this to find a lower bound for the heterochromatic number of the projective plane.
This paper presents an overview of the current state in research directions in the rainbow Ramsey theory. We list results, problems, and conjectures related to existence of rainbow arithmetic progressions in [n] and N. A general perspective on other rainbow Ramsey type problems is given.
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.
KeywordsEdge-coloring-Ramsey theory-Rainbow-Heterochromatic-Multicolored-Anti-Ramsey
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.
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 maximal planar graphs of order , we prove that a
vertex--coloring containing no rainbow faces uses at most
colors, and this is best possible. For maximal
graph embedded on the projective plane, we obtain the analogous best bound
. The main ingredients in the proofs are
classical homological tools. By considering graphs as topological spaces, we
introduce the notion of a null coloring, and prove that for any graph G a
maximal null coloring f is such that the quotient graph G/f is a forest.
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 prove that each n-vertex plane graph with girth g≥4 admits a vertex coloring with at least ⌈n/2⌉+1 colors with no rainbow face, i.e., a face in which all vertices receive distinct colors. This proves a conjecture of Ramamurthi and West. Moreover, we prove for plane graph with girth g≥5 that there is a vertex coloring with at least \(
{\left\lceil {\frac{{g - 3}}
{{g - 2}}n - \frac{{g - 7}}
{{2{\left( {g - 2} \right)}}}} \right\rceil }
\) if g is odd and \(
{\left\lceil {\frac{{g - 3}}
{{g - 2}}n - \frac{{g - 6}}
{{2{\left( {g - 2} \right)}}}} \right\rceil }
\) if g is even. The bounds are tight for all pairs of n and g with g≥4 and n≥5g/2−3.
Consider natural numbers {1,···,n} colored in three colors. We prove that if each color appears on at least (n +4 )/6 numbers then there is a three-term arithmetic progression whose elements are colored in distinct colors. This variation on the theme of Van der Waerden's theorem proves the conjecture of Jungice t al.
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 3–coloring of an abelian group G is rainbow–free if there is no 3–term arithmetic progression with its members having pairwise distinct colors. We describe the structure of rainbow–free colorings of abelian groups. This structural description proves a conjecture of Jungi´c et al. on the size of the smallest chromatic class of a rainbow–free coloring of cyclic groups. Postprint (published version)
- V Jungić
- J Nešetřil
- R Radoičić
V. Jungić, J. Nešetřil, R. Radoičić, Rainbow Ramsey Theory, Integers 5 (2) (2005) A9.
- J W P Hirschfeld
- J A Thas
J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Clarendon Press, Oxford, 1991.
Anti-Ramsey theorems, in: Infinite and Finite Sets
- P Erdős
- M Simonovits
- V Sós
P. Erdős, M. Simonovits, V. Sós, Anti-Ramsey theorems, in: Infinite and Finite Sets, in: Discrete Math., vol. 10, Colloq. Math. Soc. Janos Bolyai, 1975,
pp. 633–643.
- P Erdős
- M Simonovits
- V Sós
- Anti-Ramsey Theorems
P. Erdős, M. Simonovits, V. Sós, Anti-Ramsey theorems, in: Infinite and Finite Sets, in: Discrete Math., vol. 10, Colloq. Math. Soc. Janos Bolyai, 1975,
pp. 633-643.
Rainbow Ramsey Theory
- Jungić