# On the balanced upper chromatic number of finite projective planes

@article{Blzsik2021OnTB,
title={On the balanced upper chromatic number of finite projective planes},
author={Zolt{\'a}n L. Bl{\'a}zsik and Aart Blokhuis and {\vS}tefko Miklavi{\vc} and Zolt{\'a}n L{\'o}r{\'a}nt Nagy and Tam{\'a}s Szonyi},
journal={Discret. Math.},
year={2021},
volume={344},
pages={112266}
}
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…

## References

SHOWING 1-10 OF 23 REFERENCES
On the balanced upper chromatic number of cyclic projective planes and projective spaces
• Computer Science, Mathematics
Discret. Math.
• 2015
Borders on the balanced upper chromatic numbers of the hypergraphs arising from the n -dimensional finite space PG ( n, q) are given and constructions of rainbow-free colorings are given.
Upper chromatic number of finite projective planes
• Mathematics
• 2008
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
On the upper chromatic number of a hypergraph
• V. Voloshin
• Mathematics, Computer Science
Australas. J Comb.
• 1995
The notion of a of a hypergraph, which is a subset of vertices to be colored so that at least two vertices are of the same color, is introduced and an algorithm for computing the number of colorings of a mixed hypergraph is proposed.
The 2-Blocking Number and the Upper Chromatic Number of PG(2,q)
• Mathematics
• 2013
A 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
Saturating sets in projective planes and hypergraph covers
• Z. Nagy
• Mathematics, Computer Science
Discret. Math.
• 2018
The upper bound on the smallest possible size of the saturating set is improved to $lceil\sqrt{3q\ln{q}+1)/2\rceil$ and the connection with the transversal number of uniform multiple intersecting hypergraphs is found.
Coloring mixed hypergraphs : theory, algorithms and applications
Introduction The lower chromatic number of a hypergraph Mixed hypergraphs and the upper chromatic number Uncolorable mixed hypergraphs Uniquely colorable mixed hypergraphs $\mathcal{C}$-perfect mixed
A theorem in finite projective geometry and some applications to number theory
A point in a finite projective plane PG(2, pn), may be denoted by the symbol (Xl, X2, X3), where the coordinates x1, X2, X3 are marks of a Galois field of order pn, GF(pn). The symbol (0, 0, 0) is
A sum-product estimate in finite fields, and applications
• Mathematics
• 2003
AbstractLet A be a subset of a finite field $$F := \mathbf{Z}/q\mathbf{Z}$$ for some prime q. If $$|F|^{\delta} < |A| < |F|^{1-\delta}$$ for some δ > 0, then we prove the estimate  |A + A| +
Planar Functions and Planes of Lenz-Barlotti Class II
• Mathematics, Computer Science
Des. Codes Cryptogr.
• 1997
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, and which cannot be obtained by derivation or lifting.
An Affine Analogue of Singer's Theorem
James Singer by using the Finite Projective Geometry PG(2,p n ), proved the following theorem of the 'theory of numbers': Given an integer m≥2 of the form p n (p being a prime) we can find m+I