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Proving a conjecture on chromatic polynomials by counting the number of acyclic orientations
Citation
Dong, F., Ge, J., Gong, H., Ning, B., Ouyang, Z., & Tay, E. G. (2020). Proving a conjecture on chromatic polynomials by counting the number of acyclic orientations. Journal of Graph Theory, 96(3), 343-360. https://doi.org/10.1002/jgt.22617
Abstract
The chromatic polynomial π(πΊ,π₯) of a graph πΊ of order π can be expressed as βππ=1(β1)πβππππ₯π , where ππ is interpreted as the number of brokenβcycleβfree spanning subgraphs of πΊ with exactly π components. The parameter π(πΊ)=βππ=1(πβπ)ππ/βππ=1ππ is the mean size of a brokenβcycleβfree spanning subgraph of πΊ . In this article, we confirm and strengthen a conjecture proposed by Lundow and MarkstrΓΆm in 2006 that π(ππ)<π(πΊ)<π(πΎπ) holds for any connected graph πΊ of order π which is neither the complete graph πΎπ nor a tree ππ of order π . The most crucial step of our proof is to obtain the interpretation of all ππ 's by the number of acyclic orientations of πΊ .
Date Issued
2020
Publisher
Wiley
Journal
Journal of Graph Theory