Q-Chromatic Polynomial
The 
-chromatic polynomial, introduced by Birkhoff and Lewis (1946) and termed the "
-chromial" by Bari (1974), is an alternate form of the chromatic polynomial 
defined for graphs with chromatic number 
by
Its definition is motivated by the fact that 
for any graph with chromatic number 
, meaning dividing out the corresponding terms 
from 
provides a more compact representation than 
since 
has a smaller coefficients. When 
, 
is a polynomial of degree 
(instead of 
) for a graph with vertex count 
, and in the case where 
, it is a polynomial in 
of degree 
plus a term involving 
(Birkhoff and Lewis 1946).
REFERENCES
Bari, R. A. "Chromatically Equivalent Graphs." In Graphs and Combinatorics (Ed. R. A. Bari and F. Harary). Berlin: Springer-Verlag, pp. 186-200, 1974.
Birkhoff, G. D. and Lewis, D. C. "Chromatic Polynomials." Trans. Amer. Math. Soc. 60, 355-451, 1946.
