The Potts model and chromatic functions of graphs

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 the Stanley's symmetric bad colouring polynomial XB(G) . Moreover Sarmiento established the equivalence between U and the polychromate of Brylawski. All these functions have countable number of variables, even though the restrictions to an arbitrary graph are honest polynomials. Loebl defined the q -dichromate B_q(G,x,y) as a function of 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 certain magnetic field contribution, and conjectured that 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 the Loebl's integer partitions conjecture, and we introduce a new function B_{r,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 B_{r,q}(G,x,k) is equivalent to U -polynomial. This gives a Potts model-type formulation for the U -polynomial.