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On the balanced upper chromatic number of finite projective planes
Abstract
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.
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 .
Let be an arbitrary finite projective plane of order q. A subset S of its points is called saturating if any point outside S is collinear with a pair of points from S. Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to . The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs.
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.
With one non-trivial exception, GF(qn) contains a primitive element of arbitrary trace over GF(q).
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 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).
Let Π be an affine plane which admits a collineation τ such that the cyclic group generated by τ leaves one point (say X ) fixed, and is transitive on the set of all other points of Π. Such “cyclic affine planes” have been previously studied, especially in India, and the principal result relevant to the present discussion is the following theorem of Bose [ 2 ]: every finite Desarguesian affine plane is cyclic. The converse seems quite likely true, but no proof exists. In what follows, we shall prove several properties of cyclic affine planes which will imply that for an infinite number of values of n there is no such plane with n points on a line.
Many interesting features of sequences and functions defined on finite fields are related to the interplay between the additive and the multiplicative structure of the finite field. In this paper, we survey some of these objects which are related to difference set representations of projective planes.
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 give a systematic treatment of the theory of affine difference sets; such sets correspond to affine planes with a particularly nice type of collineation group. (This is similar to the correspondence between projective planes with a Singer group and planar difference sets.) In particular, the Desarguesian affine planes AG(2,q) admit a representation by an affine difference set; this classical result of Bose was the starting point for the area under survey. Our work includes several new proofs as well as new conditions for the general abelian case and even for non-abelian groups.
Let A be a subset of a finite field
for some
prime q. If
for some δ > 0, then we prove the estimate
for some ε = ε(δ) > 0. This is a finite field
analogue of a result of [ErS]. We then use this estimate to prove a
Szemerédi-Trotter type theorem in finite fields, and obtain a new estimate for
the Erdös distance problem in finite fields, as well as the three-dimensional
Kakeya problem in finite fields.
Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3e for every e = 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.
On affine difference sets
D. Jungnickel, On affine difference sets, Sankhya 54 (1992) 219-240.