The stipulation polynomial of a uniquely list-colorable graph

Let G be graph and let S be a set of lists of colon; at the vertices G is said to be S list-colorable if there exists a proper' /'rllnr"'Hl of G sllch that each vertexi takes its color . Alan and Tarsi! I] have shown that G is S list-colorable if and only if its graph polynomial fC(;1;..):= IT(Xi Xj) i~J does not lie in the ideal I generated by the annihilator polynomials colors available at the vertices. of the We consider the case where G is list-colorable and determine the irreducible of the remainder polynomial (or stipulation polynomial) 1c = fa mod J. We establish a bijection between the factors of and the edges of G.