Colorings of Graph Products

Abstract

In this paper, we generalize the theorem of Greenwell and Lov´asz, that
gives some graph products, that all of their colorings are by one coordinate.
We define the class of trivially power colorable graphs, that all of
the colorings of their finite powers are trivial (by one coordinate), and
give some necessary and some sufficient conditions for graphs to be trivially
power colorable. These conditions meet in a special class of graphs
called cographs. Finally we show that all colorings of the infinite powers
of trivially power colorable graphs are defined by an ultrafilter.