Patch Colorings and Rigid Colorings of the Rational n-Space

Perfect $2$-colorings of the Platonic graphs

In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

On the Defining Number of (2n - 2)-Vertex Colorings of Kn x Kn

Basis theorems for continuous n-colorings

This article is devoted to the study of continuous colorings of the n-element subsets of a Polish space. The homogeneity number hm(c) of an n-coloring c : [X]n → 2 is the least size of a family of c-homogeneous sets that covers X. An n-coloring is uncountably homogeneous if hm(c) > א0. Answering a question of B. Miller, we show that for every n > 1 there is a finite family B of continuous n-col...

A Note on the Non-Colorability Threshold of a Random Graph

In this paper we consider the problem of establishing a value r0 such that almost all random graphs with n vertices and rn edges, r > r0, are asymptotically not 3-colorable. In our approach we combine the concept of rigid legal colorings introduced by Achlioptas and Molloy with the occupancy problem for random allocations of balls into bins. Using the sharp estimates obtained by Kamath et al. o...

Extremal H-colorings of trees and 2-connected graphs

For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H. H-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H-colorings. In this paper, we prove several res...