: A k-fold coloring of a graph is a function that assigns to each vertex a set of k colors, so that the color sets assigned to adjacent Contract grant sponsor: NSFC; Contract grant number: 10671033 (to W.L.); Contract grant sponsor: National Science Foundation; Contract grant number: DMS 0302456 (to D.D.L.); Contract grant sponsor: National Science Council; Contract grant number: NSC95-2115-M-110-013-MY3 (to X.Z.). Journal of Graph Theory ᭧ 2009 Wiley Periodicals, Inc. 311