 
Summary: A note on graph colorings and graph polynomials
Noga Alon
Michael Tarsi
Abstract
It is known that the chromatic number of a graph G = (V, E) with V = {1, 2, . . . , n} exceeds k
iff the graph polynomial fG = ijE,i
of this result, using Ore's version of Haj´os' theorem. We also show that a certain weighted sum
over the proper kcolorings of a graph can be computed from its graph polynomial in a simple
manner.
1 Introduction
The graph polynomial of a graph G = (V, E) on a set V = {1, 2, . . . , n} of n vertices is fG =
ijE,i
polynomial vanishes for all values of (x1, x2, . . . , xn) S × S × . . . × S, since for each such value of
the variables xi there is some edge ij E with xi  xj = 0. This shows that the graph polynomial
encodes some information about the chromatic number of the graph, and indeed it is shown in [4]
and [1] that a graph is not kcolorable if and only if its graph polynomial lies in certain ideals. In
the first half of this note we present a new, short proof of this result, stated in Theorem 1.1 and
the two corollaries following it. The proof is based on Ore's version of the theorem of Haj´os. It is
worth mentioning that Ore noted in [5] (see also [3]) that this stronger version is sometimes crucial
for applications, but gave no examples of such an application. Our proof here is such an example,
