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).