Multicoloring the Mycielskian of Graphs

A k-fold coloring of a graph G is an assignment of k distinct colors to each vertex of G so that adjacent vertices receive no colors in common. The k-th chromatic number of G,denote by chi_k(G), is the smallest number of colors needed to give G a k-fold coloring. Let mu_p(G) denote the p-Mycielskian of G. In this paper, we show that chi_k(mu(W_n)) = 3k + [k/3] + 1 (n is even, n ges 2k + 2 > 4). Moreover, we investigate the k-th chromatic number of Mycielskians of bipartite graph. Finally, we determine the values of chi_k(mu_p(G)) (chi(G) = omega(G) = n ges 3).