Tay, Eng Guan

2020, Dong, F. M., Ge, Jun, Gong, Helin, Ning, Bo, Ouyang, Zhangdong, Tay, Eng Guan

The chromatic polynomial 𝑃(𝐺,𝑥) of a graph 𝐺 of order 𝑛 can be expressed as ∑𝑛𝑖=1(−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)=∑𝑛𝑖=1(𝑛−𝑖)𝑎𝑖/∑𝑛𝑖=1𝑎𝑖 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)<𝜖(𝐺)<𝜖(𝐾𝑛) holds for any connected graph 𝐺 of order 𝑛 which is neither the complete graph 𝐾𝑛 nor a tree 𝑇𝑛 of order 𝑛 . The most crucial step of our proof is to obtain the interpretation of all 𝑎𝑖 's by the number of acyclic orientations of 𝐺 .