Q-chromial
Alternate name
Definition
The Q-chromatic polynomial, introduced by Birkhoff and Lewis and termed the "Q-chromial" by Bari, is an alternate form of the chromatic polynomial π(x) defined for graphs with chromatic number χ>=3 by Q(u) = (π(u + 3))/(u(u + 1)(u + 2)(u + 3)). Its definition is motivated by the fact that π(0) = π(1) = π(2) = 0 for any graph with chromatic number χ>2, meaning dividing out the corresponding terms x(x - 1)(x - 2) from π(x) provides a more compact representation than π(x) since Q(u) has a smaller coefficients. When χ>3, Q(u) is a polynomial of degree n - 4 (instead of n) for a graph with vertex count n, and in the case where χ = 3, it is a polynomial in u of degree n - 4 plus a term involving u^(-1).
Related term
