Article

Rainbow-free 3-colorings of Abelian Groups

February 2012
The Electronic Journal of Combinatorics 19(1):P45

DOI:10.37236/2054

Authors:

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.

Rainbow Solutions of a Linear Equation with Coefficients in Z/pZ

{\mathbb {Z}/p\mathbb {Z}}

Article

Full-text available

Oct 2024
GRAPH COMBINATOR

Mario Huicochea

Let p be a prime,

n\in {\mathbb {Z}^+}

and

w\in (0,1)

. Given a colouring

\chi :{\mathbb {Z}/p\mathbb {Z}}\rightarrow \{1,2,\ldots ,n\}

and a linear equation

\begin{aligned} \mathcal {L}:\qquad a_1x_1+a_2x_2+\cdots +a_nx_n=b \end{aligned}

with

a_1,a_2,\ldots ,a_n\in {(\mathbb {Z}/p\mathbb {Z}})^{*}

and

b\in {\mathbb {Z}/p\mathbb {Z}}

, we denote by

R(\chi ,{\mathcal {L}})

the family of vectors

(b_1,b_2,\ldots ,b_n)\in ({\mathbb {Z}/p\mathbb {Z}})^n

suchthat

a_1b_1+a_2b_2+\cdots +a_nb_n=b

\chi ^{-1}(i)\cap \{b_1,b_2,\ldots ,b_n\}\ne \emptyset

for each

i\in \{1,2,\ldots ,n\}

. In this paper it is shown that there exists a constant

c=c(w,n)>0

with the following property: if

\min _{1\le i\le n}|\chi ^{-1}(i)|\ge wp+1\gg p^{\frac{3}{4}}

and if there exist coefficients

a_i

and

a_j

such that

a_i\not \in \{\pm a_j\}

, then

\begin{aligned} |R(\chi ,{\mathcal {L}})|\ge cp^{n-1}. \end{aligned}

Moreover, this statement is sharp in different directions. A result about the solutions of

{\mathcal {L}}

in a grid is used in its proof and it is interesting in its own right.

Rainbow Numbers of

\mathbb{Z}_p

and

\mathbb{Z}_{pq}

for $a_1x_1+a_2x_2+a_3x_3 =b

Preprint

Full-text available

May 2019

An exact r-coloring of a set S is a surjective function

c:S\to [r]

. The rainbow number of set S for equation eq is the smallest integer r such that every exact r-coloring of S contains a rainbow solution to eq. In this paper, the rainbow numbers of

\mathbb{Z}_p

, for p prime and the equation

a_1x_1 + a_2x_2 + a_3x_3 = b

are determined. Some of the rainbow numbers of

\mathbb{Z}_{pq}

, for primes p and q, are also determined.

The structure of rainbow-free colorings for linear equations on three variables in Zp

Article

Feb 2015

Let p be a prime number and Zp be the cyclic group of order p. A coloring of Zp is called rainbow-free with respect to a certain equation, if it contains no rainbow solution of the same, that is, a solution whose elements have pairwise distinct colors. In this paper we describe the structure of rainbow-free 3-colorings of Zp with respect to all linear equations on three variables. Consequently, we determine those linear equations on three variables for which every 3-coloring (with nonempty color classes) of Zp contains a rainbow solution of it.

On Perfect Balanced Rainbow-Free Colorings and Complete Colorings of Projective Spaces

Article

Jul 2024

This paper is motivated by the problem of determining the related chromatic numbers of some hypergraphs. A hypergraph

\Pi _{q}(n,k)

is defined from a projective space PG

(n-1,q)

, where the vertices are points and the hyperedges are

(k-1)

-dimensional subspaces. For the perfect balanced rainbow-free colorings, we show that

{\overline{\chi }}_{p}(\Pi _{q}(n,k))=\frac{q^n-1}{l(q-1)}

, where

k\ge \lceil \frac{n+1}{2}\rceil

and l is the smallest nontrivial factor of

\frac{q^n-1}{q-1}

. For the complete colorings, we prove that there is no complete coloring for

\Pi _{q}(n,k)

with

2\le k<n

. We also provide some results on the related chromatic numbers of subhypergraphs of

\Pi _{q}(n,k)

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).

A Rainbow Ramsey Analogue of Rado's Theorem

Article

Apr 2014
DISCRETE MATH

We present a Rainbow Ramsey version of the well-known Ramsey-type theorem of Richard Rado. We use techniques from the Geometry of Numbers. We also disprove two conjectures proposed in the literature.

Rainbow-free colorings for x+y=czx+y=cz in ZpZp

Article

Sep 2012
DISCRETE MATH

Let pp be a prime number and ZpZp be the cyclic group of order pp. A 3-coloring of ZpZp is rainbow-free for some equation if it contains no rainbow solution of the equation. In [3] Jungić et al. (2003) proved that every 3-coloring of ZpZp, with the cardinality of the smallest color class greater than four, has a rainbow solution of “almost” all linear equations in three variables in ZpZp. In this work we handle the “small” cases and give a structural description of rainbow-free colorings for the particular case of x+y=czx+y=cz, which includes the Schur equation.

Counting patterns in colored orthogonal arrays

Article

Apr 2011
DISCRETE MATH

Let S be an orthogonal array OA(d,k) and let c be an r--coloring of its ground set X. We give a combinatorial identity which relates the number of vectors in S with given color patterns under c with the cardinalities of the color classes. Several applications of the identity are considered. Among them, we show that every equitable r--coloring of the integer interval [1,n] has at least

1/2(n/r)^2+O(n)

monochromatic Schur triples. We also show that in an orthogonal array

OA(d,d-1)

, the number of monochromatic vectors of each color depends only on the number of vectors which miss that color and the cardinality of the color class.

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Rainbow-free 3-colorings of Abelian Groups

Abstract

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