Proving a conjecture on chromatic polynomials by counting the number of acyclic orientations

Proving a conjecture on chromatic polynomials by counting the number of acyclic orientations

The chromatic polynomial P(G,x) of a graph G of order n can be expressed as ∑i=1n(−1)n−iaixi, where ai is interpreted as the number of broken‐cycle‐free spanning subgraphs of G with exactly i components. The parameter ϵ(G)=∑i=1n(n−i)ai∕∑i=1nai is the mean size of a broken‐cycle‐free spanning subgrap...

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Veröffentlicht in:Journal of graph theory 2021-03, Vol.96 (3), p.343-360
Hauptverfasser: Dong, Fengming, Ge, Jun, Gong, Helin, Ning, Bo, Ouyang, Zhangdong, Tay, Eng Guan
Format: Artikel
Sprache:eng
Schlagworte:
acyclic orientation
chromatic polynomial
combinatorial interpretation
graph
Graph theory
Polynomials
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Zusammenfassung:The chromatic polynomial P(G,x) of a graph G of order n can be expressed as ∑i=1n(−1)n−iaixi, where ai is interpreted as the number of broken‐cycle‐free spanning subgraphs of G with exactly i components. The parameter ϵ(G)=∑i=1n(n−i)ai∕∑i=1nai is the mean size of a broken‐cycle‐free spanning subgraph of G. In this article, we confirm and strengthen a conjecture proposed by Lundow and Markström in 2006 that ϵ(Tn)
ISSN:0364-9024
1097-0118
DOI:10.1002/jgt.22617