Multicoloring and Mycielski construction

The generalized Mycielskians of graphs (also known as cones over graphs) are the natural generalization of the Mycielskians of graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer p>=0, one can transform G into a new graph @m"p(G), the p-Mycielskian of G. In this paper, we study the kth chromatic numbers @g"k of Mycielskians and generalized Mycielskians of graphs. We show that @g"k(G)+1= =1, p>=0 and n>=2. Finally, we prove that if a graph G is a/b-colorable then the p-Mycielskian of G, @m"p(G), is (at+b^p^+^1)/bt-colorable, where t=@?"i"="0^p(a-b)^ib^p^-^i. And thus obtain graphs G with m(G) grows exponentially with the order of G, where m(G) is the minimal denominator of a a/b-coloring of G with @g"f(G)=a/b.