Broken wheels are SLC. (English) Zbl 0597.05033

Read, Ronald C.

Ars Comb. 21A, 123-128 (1986).

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

Keywords:

strong logarithmic concavity; chromatic polynomials; broken wheels; SLC