The Potts Model and Chromatic Functions of Graphs

Abstract. The U -polynomial of Noble and Welsh is known to have intimate connections with the Potts model as well as with several important graph polynomials. For each graph G, U(G) is equivalent to Stanley’s symmetric bad colouring polynomial XB(G). Moreover Sarmiento established the equivalence between U and the polychromate of Brylawski. Loebl defined the q-dichromate Bq(G, x, y) as a function of a graph G and three independent variables q, x, y, proved that it is equal to the partition function of the Potts model with variable number of states and with a certain external field contribution, and conjectured that the q-dichromate is equivalent to the U -polynomial. He also proposed a stronger conjecture on integer partitions. The aim of this paper is two-fold. We present a construction disproving Loebl’s integer partitions conjecture, and we introduce a new function Br,q(G; x, k) which is also equal to the partition function of the Potts model with variable number of states and with a (different) external field contribution, and we show that Br,q(G;x, k) is equivalent to the U -polynomial and to Stanley’s symmetric bad colouring polynomial.