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 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.