Rainbow-free 3-colorings in abelian groups

Rainbow-free 3-colorings of Abelian Groups

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 g...

THE STRUCTURE OF RAINBOW-FREE COLORINGS FOR LINEAR EQUATIONS ON THREE VARIABLES IN Zp

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 equation, 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. Consequen...

How many delta-matroids are there?

J. Makowsky and B. Zilber (2004) showed that many variations of graph colorings, called CP-colorings in the sequel, give rise to graph polynomials. This is true in particular for harmonious colorings, convex colorings, mcct-colorings, and rainbow colorings, and many more. N. Linial (1986) showed that the chromatic polynomial χ(G;X) is #P-hard to evaluate for all but three values X = 0, 1, 2, wh...

On the complexity of generalized chromatic polynomials

J. Makowsky and B. Zilber (2004) showed that many variations of graph colorings, called CP-colorings in the sequel, give rise to graph polynomials. This is true in particular for harmonious colorings, convex colorings, mcct-colorings, and rainbow colorings, and many more. N. Linial (1986) showed that the chromatic polynomial χ(G;X) is #P-hard to evaluate for all but three values X = 0, 1, 2, wh...

Colorings Of Plane Graphs With No Rainbow Faces