A k-fold x-coloring of a graph is an assignment of (at least) k distinct
colors from the set {1, 2, ..., x} to each vertex such that any two adjacent
vertices are assigned disjoint sets of colors. The smallest number x such that
G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by
\chi_k(G). We determine the exact value of this parameter when G is a web or an
antiweb. Our results generalize the known corresponding results for odd cycles
and imply necessary and sufficient conditions under which \chi_k(G) attains its
lower and upper bounds based on the clique, the fractional chromatic and the
chromatic numbers. Additionally, we extend the concept of \chi-critical graphs
to \chi_k-critical graphs. We identify the webs and antiwebs having this
property, for every integer k <= 1.Comment: A short version of this paper was presented at the Simp\'osio
Brasileiro de Pesquisa Operacional, Brazil, 201