  A proof of the Generalized Road Coloring Problem, independent of the recent
work by Beal and Perrin, is presented, using both semigroup methods and
Trakhtman's algorithm. Algebraic properties of periodic, strongly connected
digraphs are studied in the semigroup context. An algebraic condition which
characterizes periodic, strongly connected digraphs is determined in the
context of periodic Markov chains.
