 
Summary: Bounds on the Chromatic Polynomial and on the
Number of Acyclic Orientations of a Graph
Nabil Kahale 1 , Leonard J. Schulman 2
Abstract
An upper bound is given on the number of acyclic orientations of a graph,
in terms of the spectrum of its Laplacian. It is shown that this improves upon
the previously known bound, which depended on the degree sequence of the
graph. Estimates on the new bound are provided.
A lower bound on the number of acyclic orientations of a graph is given,
with the help of the probabilistic method. This argument can take advantage
of structural properties of the graph: it is shown how to obtain stronger
bounds for smalldegree graphs of girth at least five, than are possible for
arbitrary graphs. A simpler proof of the known lower bound for arbitrary
graphs is also obtained.
Both the upper and lower bounds are shown to extend to the general
problem of bounding the chromatic polynomial from above and below along
the negative real axis.
1 XEROX Palo Alto Research Center, 3333 Coyote Hill Road, CA 94304. Partially supported
by the NSF under grant CCR9404113. Most of this research was done while the author was at
the Massachusetts Institute of Technology, and was supported by the Defense Advanced Research
