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Broken wheels are SLC. (English) Zbl 0597.05033
A polynomial $$a_ n\lambda^ n-a_{n-1}\lambda^{n-1}+...\pm a_ 0$$ with coefficients of alternating sign is said to have the property of strong logarithmic concavity (SLC) iff $$a^ 2_ r-a_{r-1}a_{r+1}>0$$ for $$2\leq r\leq n-1$$. It has been conjectured that the chromatic polynomial of any graph is SLC. In particular, D. Gernert [Methods Oper. Res. 51, 307-314 (1984; Zbl 0546.05028)] conjectured that the chromatic polynomials of broken wheels are SLC (a broken wheel is a wheel in which some ”spoke” edges have been deleted). The author proves the latter conjecture.
Reviewer: R.C.Entringer

##### MSC:
 05C15 Coloring of graphs and hypergraphs