Abstract
A k-fold n-coloring of G is a mapping φ: V(G) → Zk(n) where Zk(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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
