Abstract
A k-fold n-coloring of G is a mapping φ: V(G) → Z k (n) where Z k (n) is the collection of all k-subsets of {1, 2,..., n} such that φ(u) ∩ φ(v) = \( \not 0 \) if uv ∈ E(G). If G has a k-fold n-coloring, i.e., G is k-fold n-colorable. Let the smallest integer n such that G is k-fold n-colorable be the k-th chromatic number, denoted by χk (G). In this paper, we show that any outerplanar graph is k-fold 2k-colorable or k-fold χk (C*)-colorable, where C* is a shortest odd cycle of G. Moreover, we investigate that every planar graph with odd girth at least 10k − 9 (k ⩾ 3) can be k-fold (2k + 1)-colorable.
Similar content being viewed by others
References
Borodin O V. A structural theorem on plane graphs and its application to coloring. Diskret Matem, 1992, 4: 60–65
Klostermeyer W, Zhang C Q. 2 + ε-Coloring planar graphs with large odd girth. J Graph Theory, 2000, 33: 109–119
Klostermeyer W, Zhang C Q. n-Tuple coloring of planar graphs with large odd girth. Graphs Combin, 2002, 18: 119–132
Lin W. Multicoloring and Mycielski constructions. Discrete Math, 2008, 308: 3565–3573
Scheinerman E R, Ullman D H. Fractional Graph Theory. In: Wiley-Interscience Series in Discrete Mathematics and Optimization. New York: John Wiley & Sons, 1997
Stahl S. n-Tuple colorings and associated graphs. J Combin Theory Ser B, 1976, 20: 185–203
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ren, G., Bu, Y. k-fold coloring of planar graphs. Sci. China Math. 53, 2791–2800 (2010). https://doi.org/10.1007/s11425-010-3083-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-010-3083-y