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New expressions for order polynomials and chromatic polynomials
Citation
Dong, F. (2019). New expressions for order polynomials and chromatic polynomials. Journal of Graph Theory, 94(1), 30-58. https://doi.org/10.1002/jgt.22505
Abstract
Let πΊ=(π,πΈ) be a simple graph with π={1,2,β¦,π} and π(πΊ,π₯) be its chromatic polynomial. For an ordering π=(π£1,π£2,β¦,π£π) of elements of π , let πΏπΊ(π) be the number of integers π , where 1β€πβ€πβ1 , with either π£π<π£π+1 or π£ππ£π+1βπΈ . Let π²(πΊ) be the set of subsets {π,π,π} of π , where π<π<π , which induces a subgraph of πΊ with ππ as its only edge. We show that π²(πΊ) =β
if and only if (β1)ππ(πΊ,βπ₯)=βπ(π₯+πΏπΊ(π)π) , where the sum runs over all π! orderings π of π . To prove this result, we establish an analogous result on order polynomials of posets and apply Stanley's work on the relation between chromatic polynomials and order polynomials.
Date Issued
2019
Publisher
Wiley
Journal
Journal of Graph Theory
DOI
10.1002/jgt.22505
Description
This is the final draft, after peer-review, of a manuscript published in Journal of Graph Theory. The published version is available online at https://doi.org/10.1002/jgt.22505